Dome 2

Ffr {o,o,O} Not e: AS..L12 fl Each vert ex o n the PPT is then trunslated on to t he surface of th e c ircu mscribe d sphe re alo ng a lin e pil ssing t h rough th e res pec t ive ve rtex and t he o rig in {a, 0, 0 1of th e po lyhedron. The ele men ts connect ing the t ra nsl at ed ver te x fo rm the chords o f a 3 ,waY gr e at c ircula r grid. Each poi nt of subdivisions is then con nec t ed with line segments perpendi cular to their respective princ iple side thus gi ving a 3-way gri d c o mpr ise d of equ l lat er al and right triangl es . A I Note : AH' 12 B (0, 0,0) '----r The po int s o f su bd ivision o n each pr in c ipl e side of th e PPT are connec te d with line segments paral lel to the ir respect ive sides . Each line segme nt inte rsec ts a t a n umbe r of po int s wh ich defi ne a gr id of subdi vision. Du e to t he meth od of su bdi vision, smal l equ iluteral tr ianqu lar "wi ndows" occur In th e grid . Method 2: {This met hod produces equal divisions along ttlf! spherical PPTand results in, for example, 3 different triangles in the 3 with 3 strut l engths. MNhod I has 2 triangles and 3 st r ut s] The PPT is subdivided into n fr eque ncy with the part s c hose n as equal ar c divisions of th e ce nt ra l angles of the polyhedron. A .--._ _ ..",...-..,--,-,.-." A Me t hod 7 : Go AT / TI A f!, No t e : Aa- ,.t.12 Smal l triangul ar windows occu r 1; 2. Note : AT 1- IT I AL I I 1 I I Ie. ( 0,0,0) z, No t e : ABC is a rig ht tr ia ngl e c. A'/ I =:,v= I "'>.iC I ")"J A '/ I '>J I '>J t '>J, he PPT is su bd ivi ded in to n frequenc y with t he ports chosen 05 fJ qUd l arc d ivis io ns o f the [ The chord factors and o t her data WI'? publis hed las t under the name " Triecon " we r e develo p ed from this method. I n qener«! the t ris con breakdown (Class II ) i s better for l urge do mes because th e number o f differ en t strut long t hs increases arithmetical ly wi th the t r iscon (i.e ., 6v has 6 d itteront st rut l engths, 8v tms 8, 12v hilS 12 etc .I en d geo merricd/l y wi t h the ette rnete (Class II_ Fo r smelt frequency domes the diitercnce is no t t ha t si qniticent.] Thi s me t hod is so met imes ref erred to as t he regu lar t ria contuhedr al qeodesic grid a nd WaSdevelo pe d by Duncan Stu art. The PPT may be de scribed as six right t r ian gles each be ing a re flec tion or rotati on of the other, Method 3: 1 8 • (O, O, O) . To comple te t he 3 ·way grid connect a lte rnate po int s of su bd ivis io n of side AB t o t he poi nt s of div isi on of side BC. - e: " 4' The po ints of d ivisio n on th e side CB wer e fo rmed by ex tendi nq a line t hro ugh t he po in ts of subdivi si o n on side AC perpendi c ul ar t o side CB. H · . d h . f hdivi avrnq acqUire t E: oornt s 0 su IVI Slon ' __ a lo ng th e three sides of th e t riangle , 12 i 34 d iagona ls ar e d ra wn fr o m e ac h poin t o n side .....-1 6 AC to a lte rnat e points of sides At! and BC. I'he cente rs o t these "windows" are fo und and are use d 7\ as the verti ces of a 3-way (Jrid fo r the PPT. The ve rt ices are t he n t ra nsla ted o nto t he surface o f the c irc u mscribed sphe re alon q a l ine passin g t hrough t he respect ive vert ex 1. and th e o r igin (0 , 0 , OJ of t he pol y he d ro n , The ele ment s jo ining t he tran slated vertices fo rm the c hor ds o f a ,3 -way gr e at ci rele grid. ;ftA\ Throu gh ro tati o ns an d refl e ct io ns of the ba sic u nit a nd it s subdivis io ns . t he en ti re d1 3-way gri dd ing ot t he PPT may be fo und. ), >1' !' "'I .,.J, A e ' "'""-"'" (0/ 0, 0) The vertices of t he 3 ,wa y grid ar e the n t ra ns lated to t he su r face of th e c irc umscri bed sp here a long a lin e passi ng t hr ough t he respect ive ve rt e x and t he o rig in 10, 0, 0) of th e pol v hedr o.i , The elements jo in ing t he t ra nsla te d vertices fo r m t he c hor ds of a 3 ·way gma t c irc le gri d. Method 4: he poi nt s of subdivision o n each princ ipl e side of t he PPT me c o nnected with linc seg me nts simila r t o method 1. However , the line segme nt s a re not pe rpe nd icu lar to their resp ective sides. Up o n comple t ion o f the co nnections i:l gr id is created. Du e to th e metho d o f subdi vis ion, small t rianqu lar "windows" occur in the grid . A [ This met bo d /s baS/Call y rhe Sdme ilS method 3 excem t fl at if/stedd o f d ivi d ing si de AB of th e ri ght triangle wit h the eq(Jal arc di vision s, si de A Cis divided. The resr of tht:.· procedure is the samt!. give" the new sl iJr t iny p ain r.1 - In t his me thod o f subd ivis io n we Shol l t reat onl v t ri ang le ABC. The re ma ini ng sect io n f the PPT ma y be found th rough ro t at io ns and refl ect ions of thi s ba sic Thi s is Yz S true of a ll met hods. The li ne AB is subdivide d into part s c hosen as equ al arc d ivisio ns of the centra l an nie o f t he (go, o; po lyhedron. Once th e su bd ivisi o ns ar e fou nd they are used t o find the poi nt s of di visi on on side Ar: an d CB. Perpendi cul ar s thr ough th e points of d ivis io n on side AB are ex te nded t o side AC , th is giving th e point s ot su bd ivision on side AC Note : Nof&: A Nota: Al .= W Nore : alvldl e do (Jol- /Ie on ifle same-J I13t1 f circte f'ldll7e-. 13 3 Nofe-: "..., Aa.. = a.,.1 3c- ::k- f, /Vue : 1h e- f re'fuenc:J Increases Jeomefn cal!J A A where cf .= chord facto r cf .= cent ra l angle c B , A No te : A8 IS pata lle] to 12 Ao I ab Win dows are equrl ntur al uiauqles c. cf 2 (sin 6 / 2 e> l? £ @ .--' -v ;; ".,p,:",.k':"",,- - __J __- J whu-c- <f : t lF'"- ( OA O) Th e ce nters of t hese "wind ows" ar e found 011 t he pla ne of t h,' PPT an d .m' used e Ih e ve rti ce s of t he 3· way ar id l eir tho PPT. They a re th e n t r anslatc d o n t o t he su rface o f t he ci rcumscr ibed sphere .lonq "Iinr. passinq t hr uuqh t he res pec t ive vertux and t he origin (0 , 0, 0 ) 01 t h, ! polvhedr on , Tho c lements connecti ng t he ttonslu tud vertices form th e cho rds of a 3,way qrca t circular grid. Th e in te rior po int s ar e f ound by passing great cir cle a rcs through t he pr ev iousl y found mi d po int s of each pr in cipl e side and fi nding t he mid points o f each sid« of each ne w t riangl e ill the sa me manner as above. 8y i.e., th e sphe rical polyhedra l t r ia nql c is subd ivi ded in to a low f roqu encv su bd ivision, i. e, 2.r , with pa rts c hosen as eq ual a rc div isio ns of th e ce nt ral angle of the po ly hedron. 1v Method 4 : Thi s method is an a lte ra tio n o f met hod s 1 3 a llo wing lor t ru ncat io n with in the eq ua to r ial zo ne ot t il e sphe r ic al form. It is deve loped with lesser c ircle as well as gre at ci rc le arcs so th at t ru nca tio n ma v be clone wi t hou t requirinq specia l e leme nts . A se t of pa rall el pla nes, fal ling in th e equatorial region, a re prov ided through t he geode sic sphere, pe rpendicu lar to any give n pola r axi s. Due to t he le ss sy mme tr ical charac te ris t ics o f t h is me th o rl it is used primari ly fo r srna ll fre q ue ncy st ruc tu res . Th e n umber of re lati ve di ffe renc es j n ed ge le ngt hs a l'e gr eat er th an any of t he other me t hods. Cr.ASS II Method 1: The PPT is subdivi de d into n fr equency , wit h th e parts c hose n as eq ua l di visio ns a long t he t hr ee pr inc iple sides. For furt her suboiv isons each new tr iangle is sub d ivided as in th e prev ious ste ps and connected to c o mple te t he 3-WdY gr id. By kn owlnq th e ce nt ral an gl es (S ) t he chord facto rs may be ca lcu lated by t he following equa tion : Each poi nt is the n connected wi t h gr e at circle a rcs t o com ple te t he 3· way grid. Mc tho d 3: Th is me t hod is someti mes referred t o as t he alternate qeodeslc qrid. Usuall y , i t is developed by starti ng with a small f requencv and th en su bd ividing further to t he desi red fr equency by fol lowi ng a ge ometr ica l pro gress ion as per exa mple : ,," will de up, .Ip wit h i n9 be l ore do mes I I tr ianqlos. ound to 4 'ater ru n Adju st i alves best i> . 11 A \3) Not e: 'AB .L 12 Sma ll trianqular windows occur r z, Not e: ATI- 12 Not e : AT f IT A/ I I I I I I (oo.o) c:.. (a ao) D ?, No te : ABC is a r ight tr iangl e arts c hosen a. (!q ultl arc di vis io ns of \ c A/" I ,:-v ,\ '::"J,/ I """"l /j5 S Each ve rt e x on th e PPT is th e n t run slated onto t he su rface of the circ umscri be d sphere alo nq a lin e passinq th rou gh the res pect ive ve rt e x an d th e or igi n (0. 0, 0) of th e pol vn eor on. The e le ments connect inq th e t ranslate d vert ex lorm the chords o f a 3 ' WilY grea \ circular grid . A ""--= " '-I I "J I>-J To co mplete t he 3 ,waY gri d co nnect alt erna te po ints o f su bd ivis ion of side AS t o the po int s o f d ivisi o n of side BC. - [The chord factors en d other dam we p Uhli " h" d 1",1f year un der t he na me "Triocon" wer e devel oped fr om t his met ho d. I n qenoret [h e triacon breakdown (Class 1/ ) is better for large domes beceuse t he numher of ditier ent strut leng rhs i ncr eases arithmetically wit h the t r i sco n (i.«.• 6v hDS 6 a i tteren: st rut len gt hs. Bv has 8. 12v hn« 12 etc. } en d qeo me tricettv with the eltemote [Clnss I), For small f re q uency domes th e difference is II Ot rh at si onitrcant.] Ihi s me thod is somet imes re fe r red to as the requla r t riacontahe d ra l ueodesic grid a nd WaS devel oped by Du ncan Stuar t . The PPT ma y he descr ibed as si x right triangles eac h be ing a re flec ti on or rot at ion of t he ot her. Note : AB...L U Each point o f subdivi sions is then co nnec t ed wit h line segments perpendi cular t o their respe ctive principle side thus givi ng a 3 ·wa y grid co mprised of equil ateral and right triangl es. A Me t ho d 3: Met hod 2: Til e pr T is subdi vided in to nl rnquencv wit h th rho (J f l hl! polvnpdrun. \f} G The po in ts of subdi vision on cac n princi ple side o f the PPT are connecte d with line segme nts simi lar t o met ho d 1. However, rhc line se gm ents a re not perpendi c ul ar to th cir res pect ive si des . Upon co mpletion of t he co nnections cl grid is cre ate d. Due t o lhi' me thod of subd ivis ion, sma ll u ianqula r " wind ows" occur in th e gr id. The centers of these "windows" are found an d are used as the vert ices of <J 3 ·way gri d fo r t he PPT. The ve r t ices are t hen t ra nslat ed onto t he surface of the ci rcu mscribe d sph er e alonq a line passing t hrough t he resp ec tive vert e x an d the o ri gin (0, 0, 0) o f the poly he dro n . The e lemen ts joi ni ng th e t re nslate d vert ices form t he chords of iI 3·way ureat circle gr id. { Thi s met hod IS oesics tt» the seme as method 3 excep t t hat instea d of di vidino side AS of the right t ria n yl e wit h t he eoue! arc: d i visions, si de AC is dt vtdcd. The rus t of ttre procedur e IS t h« same, gI ven the 1l0W s/dr t lng po;m.l I \:, . '\ " .-. " I \! :> Through ro t at ions an d refl ect ions of th e '\ " , ,\> // bas ic un it and its subdivi si o ns t he e nti re J.>"t' ,' I ;;., 3·way gfldd mg o f t he PPT may be fo und. J. "' 1 1 ' A P ,. ' '0=' ' ..... (0/0, 0) ._- ..a The ver t ices of t he 3·way gr id are t he n t ransl ated to th e surface o f the circu mscr ibed alonq a line piJssing t h roug h th e resp ect ive vertex and th e or igin (0, 0, 0) of t he potvhedro.i. The e leme nts jo ining t he t ranslated vert ices fo rm the chords of iJ 3 ' waY great circ le gr id. Method 4 : Nof&: AI A * .»: No te: Al ' 12 Noie : a./u,d/ e do /lof he on the same-g t13l1. f circle- 'p!?lI1e.. 3 13 rtote: ; ,...., Aa.= a-I (Vote. ; The. fre1uenc:r meretlsesyeomefrtc<j!y A A where cf < chor d fact or J ' cent ml angle c B fI ( ' I i j A No te : A B is p," Ril l!1 t o 12 Ai'. -t :)h Wlrldcm are etllil liltor'll l Not e : A 11 12 ( 0,0,0) c. with th e parts ch osen as equ al ar c di visions of the cf 2 (sin 6 / 2 fy h) th e cho rd facto rs may be Calculated by the A b) W· I'J -n connected with qrc at rnplete the 3 ''1'1 <1,/ grid. 8v Idivision on pl'inciple side of t he PPT ar e co nnecte d wit h linc seqments ra, pective side>. line segme nt inte rsec ts at a numbe r of poi nt s wh ich division. DOa td th e method of subdivision, small equilatera l trianqut ar r In the griel. pherica] polvhedral t rian gle is subd ivide d into a low freque ncy subdi visio n, 'jth part s chosen as eoua l arc d ivi sio ns of th e cent ral ang le o f t he po lyhe dr on. metimes referred t o as the a lte rnate geodes ic gri d . Usuallv, it is ting with a \n1<11I freq ue ncy and t he n subdividi ng further to th o mcv by foll owi uq u geome trica l prog res si on as pe r exa mple: -: ..... I Of - I I Z diVided int o n ' reQlJllncy , ....1'1:1 chlMn as equal alonq ;lple sides. i.... each new II iangl e liIiiIiiidIcI_t 'n It," previ ous .l 'l pSand eornolete the :3 v/lIy gr id. alterati on (,I me t hod s 1·3 allowinq for t ru ncat ion wi thi n th e I of the soherlcsl form. It is deve lop ed wit h lesser circl e as we ll ClS ' so that truncation may be done withou t req uir ing specia l cle men ts. 1,,1planes, fall it' y In the equato rial regi on, are pr ovided t h rough t he ".11.. perpendic ular t o onv give n polar axis . Due to t he less sy mme t r ica l of thrs meth od n is used prima rily fo r sma ll f reque ncy struc t ures . "I relat ive in ed ge lengt hs are greate r t han anv of t he ot her fz ..."':'... .......,- . " . and vert e x. ,.,..., ... ........ n face whe re: V = no. of verti ces } F = no. of faces for total spher e E '" no. of edges ....... -- V frequ ency of su bdivision / .-, icosahedr on edge V "' F - 20."..2 E ,. 'OvL E'; Each vert ex o n th e PPT is then t rans lated l.. alo ng a line pa ssing t hrou gh t he o rigin (0, 0, 0) of th e polyhedron and it s res pecti ve vertex , on to t he su rface of the cir cu mscri bed sphe re . Th e e leme nt connecting t he translated vertices form t he ch ords o f a 3·way great circul ar grid. (0,0,0) )fjy'2. wh er e: 11 = -z:- V no. of verti ceS} F = no. o f face s for t otal rhombi c tr iac ontahedral spher e E '" no. o f edges V = frequency o f SUbdivisi on - demOllSt rat es sy mmet ries as illustr at ed in the following figu re', V =iL+2. F = 2 (it) E = 3 (fl.) edge tJok.- ' AI =' 1"2- 4 ABC I S a f:jt/1/:rfa70-1t:>. Th e PPT is su bdivide d int o n fr equency , with the pa rts chosen Cl S equal d ivisions along t he three princip le si de s. c A Each point of SUbd ivisio n is t he n co nnec ted with a line segment parall el to th ei- re spect ive sides th er eb y giving a 3-wa y gr id so th at a ser iex of equ ila te ra l tri ang les are formed. , 1\ ( A • ; icos ahedron edge Class II : - based on th e qu asi-regular po ly he d ra l forms, mos t generally the rh ombi c triacon tah ad ron . - frequency of subdivi si on ma y o nly be even. METHODS OF GENERATINGg-WA.... GEODESICGRIDS Up on using t he sphe rica l fo rrn as a str uct urel u nit , it is ap pa ren t t hat th e basic po lyhedra l f or m, ill it s fundame ntal st ate, can IIOt sotis f y th e range o f condi tions t ha t IT'US! be geometr ica lly and st ruc tu rally met . Th ere have been many metho ds developed fo r reduc ing t he basic po lyhed ra l form int o 0 larger nu m ber of components f rom wh ich the geo metrica l pr operti es may be made to remain wit hin t he struct u ral fabri ca t io n and erection Iimi ts for a desir ed confi gu rati on , Several met ho d s of qen erat inq 3-way geodesic gri ds are d iscussed he re in a broad se nse to give t he experi me n tel' a hasi s from whi ch ot her methods may be devel oped. Th e met hod s d esc r ibed her e may be cons ide red as having characte rist ics of one of the two foll owing cl ass ific ations: Class I: - based on regu lar polyh ed ra l for ms , most gen er al ly th e icosahedro n. - frequency of su bd ivis ion may be odd or even. Ori entati on = th e ori entat ion th e po ly he dr al form ha s in space wit h respect t o t he o bserve r. Three orie nta t ions are co nsi dered: Note : AB is par all el to 12 Due to t he sy mme t r ica l characteristics of th e basic polyhedral form o nl y one face. o r po rt io ns of one face, of the pol yhedr on is used for ca lculat ing th e geometri ca l pr operti es of the st r uct ura l configurati on. The remaining faces may be found by rotati o ns an d /o r ref lec t ions of th is princ ipl e polyhedral tr iangle and it s t ra nsfo r ma tions . CLASS I Method 1: { This me t hod i s wh at was publi.\hed i ll Dornebook One under th e name " alternate " ; and is th e geometry of th e Pacifi c Dame, Alumi num Sun Dome, Pil l ow Dome, etc.] " \' \ ' \' cf = 2 2 r.c. Math ?nnap/e.- p(l/rtJecJrttl TrI"Mjle- \....- (o 0, y.0 _ --" rnnOIEI-e slide- The following sect io n is an or ticto by Jo e Cl inton on the different methods of producing geo desics from the ic osahedron. The tech nique he uses in vol ves analytiml geometry (Fu ller used spheric el trigonometry ) and th e cal culations are done with .. computer , The general p rocedur e was co find tho 3-di mensional coor di na tes of th e vertices of the grid On the sphericet surface, using the ditterem' methods, and th en to calc ulate the chord l engths, angles etc . with these coor dine tes and anal ytical torrnutss . Joe worked wi t h Fuller on his progra ms and was fu nded bv NASA on a proiect called "Struc tu re! Concep ts for Fut ure Space Mi ssions" . The specific motivation for devel oping th ese methods was to ha ve a variety of forms to combine in large space f ram e dome s. For example the Expo dome in Montr eal i s a combination o f a 32-frequencv regul ar tri econ (Cl ass II, me t ho d 3) and a 16· fr equency tr un cstshte alternate (Class I, m ettiod 3). Wirh k no wn pa rameter s and sophist ica ted ana lvs is. l argo str uc t ures can b« opt i mized IJv aittsr en t co mhinntions and differen t me thods; trowever , f or sma ll st r uc t ur es (up to 40') tnov are not generallV t elcvent, ItWMt we gcnC'(dllv colt "st terns tc" breakdo wn in this book t end i n Dornebook Orvc) . , J Of1 classifies ;JS "Ciuss I ", what we c,'JI' " triecon ", he clessltlc s as " Class 1/" . Joe. wr ote this sec t ion md i fl lv witt: the i n ten t o f cornrnunicetinq t he st ate o f devetoomenr of geodesic qcome tries and th e hope ttuu it would be an ai d to those i n terested i n ,,'xploring and ex pan ding t his fiel d. Dotuebook comments arc in i talics. , " \ Joseph D. Cl in to n BASIC DEFIN1TlONS Axi al angle (o mp.gaJ2 ) "-' all an \lle fo r me d by an e lemen t and a rad ius from the ce nte r of th e polyhedron meeti ng in a common poi nt. The vertex of t he axia: angle is chosen as t hat poin t common to t he polyhed ro n element and rad ius. The ax ial angle.(lmay be f ound if t he centra l is known by t he fol lowing equati o n : = 180- £ 2 6 180 - (1l.. ,d 11. ) :axltL/4.il ( 0 a 0 ) / , . - _. 1!" cent(}y- pOI"hed/'7lrr .--Jl K . Centra l angle [delta b ) an angle formed by two rad ii o f t he po ly hedron passin g t hrough the end poi nt s of an el emen t o f the polyhed ron. Th e vertex of th e ce nt ral angle is chose n as that point co mmon t o both radii (the ce nter of the pol yhedron ). Th e ce nt ral angle b ma y be found by kn owing th e ax ial angles a , &.a,. at ea ch end of an el e me nt. Th e length of any eleme nt for larger struct u res may be found by t he eq ua t ion : 1= cf x r wh er e: r th e rad ius o f the desired struc t u ra l f orm I = th e length o f the new eleme nt Dihedral an gle (beta;8 ) '" an angl e formed by two pl anes meet ing in a co mmon line . The tw o planes th em selves are faces of t he dihedral angle, and th e e leme nt is th e co mmon line. To me asure th e di hedral angle measure th e angle wh os e vert ex is o n the eleme nt of the dihedral an gle and whose sides are perpendicul ar to th e element and lie on e in each face o f t he dihedral angle. of the dfhedrM Face angl e ( alpha 0< } = an angle fo rmed by two ele ments meet ing in a common point e /ertff'l-r/,tf ffie, and ly ing in a plane th at is one of the o{he ral f aces of th e polyhed ron. fJ.(f, ()f drhedral & 10il Fr equency INu Y I = the number o f parts or segme nts in to whi ch a pri nc ip le side is su bdi vide d. Pr incipl e side (PS) = anyone o f th e sides of th e pri nc ip le pol yh edrol t riangl e, Face =any of the pla ne po lygo ns making LIp the surface of th e struc tu ra l form. Chord factor (cf ) > the element len gths ca lculate d ba sed o n a rad ius o f a non-dimensi on al unit of o ne for th e spherical fo rm with the end point s of t he ele ments coi nci de nt with th e su rfa ce o f th e sphere . If the cent ral angle b is kn own th e cho rd fa ct or may be cal culated as follows: Princ iple po lyh edral tr iangl e (PPT) =an y one of t he plane equ ilate ral tri angles wh ich fOIT'l th e fac es of th e regular polyhedr on. .... MAA AA1A zr ?v 4- V ZV qv (£jl' 6,0 vert BX idtmt if icat ion diagra m lor Class I i"alte "na t" '" brca kdown . Draqrnm is one [ace of icos ahe dron. 1\ cho rd factor is a pur e nu mbe r whi ch, wh en multipli ed by a radi us, gives a str ut length. We o btained th e cho rd fac tors we pri nted in Oomebook One fro m various sou rce s. Since t hat time, we have made coni act wi t h J oseph D. C1i nt on wh o worked o ut th e glJodesi c co mpu ter pr ogr ams fo r NASA. Whnt we ca lled "alte rnate" breakdown in Domebook One, and in mu ch uf thi s book is cl assified in the computer pr oqr arns as CLASS I. What we call "tr iaco n" is CLASS II. In each class th er e are several methods, as described by Clin to n u n pp. 106·107. On t he pa ge o pposite we are pri nting Cla ss I, method 1- "alternat e", th e math emat ics we have used for most of o ur domes. On p. 110, we are pr inting Class II, met hod 2, t he math we hav e used in o u r "rriacon" domes. On p. 112 we are printi ng so me alte rna t ive methods of both classes , and so me t et rahedron and oc t ahedro n chord fact ors. It you want to ex plore ful ly t he d iff erent methods we sugges t t hat you make mo dels and st udy Cl inton ' s geodesic mathemati cs sec tio n. Figur es o n the o ppos ite page ar e for icosahed ro n based Class I geo desic spheres as dis cover ed by R. Buckminster Ful ler. Th e numbers are from comput er readout generated by proqrarns deve loped by Joseph O. Clinton under a NASA-sponsored researc h grant, "Advanced Structural Design Concep ts For Future Space Missions," Final Report. March, 1970, NASA Contrac t NGR 14-008-002. Figures given do not refer to structural strength. Th e higher t he fr equency, t he f latter tne angle of th e dome's faces (th e " d ihedra l" below) and the more cri t ical Is accu rat e workmansh ip to prevent "popping in" of a vertex unde r load. Big domes usually are made fr om folded pla tes whi ch give the skin a cross sect ion, o r th ey are mad e fro m two domes o f the same or differe nt frequenci es but d iff er ent size , one ins ide the o t her and laced t oget her with addi t ional str uts, as at EXPO ' 67. An icosahed ro n has t went y ide n tical equ ilatera l t riangle fac es and twelve pentaqo nal vertices. The diagram here is one face o f the basic icosahed ron (d ivide d int o t he number o f additi onal faces as requ ired by the des ired frequency. ) Th e d ifferent lengt hs o f ed ges of these additio na ! faces cause t he icosa face t o assum e a more spherical shap e. For a given d iameter, the highe r t he frequen cy , th e mor e sp here-like th e icosahedron becomes. Th e numbers on the d iagram are vertex identi fications, an d are ref err ed to in the tables start ing with 0,0. Th e foll owing is an explana t ion o f the t abl es , and iill examples given ar e fo r a "two frequ ency" breakdown. Note th at t he diagram, regardl ess of what port ion o f it is used , does not imply size. Starting wi th th e two fr equency sect ion: 2·FREQUENCY ICOSAHEDRON means a 2-frequency br eakdown of a basic icusa. Lo oking at Fig. I, we see that for a 2-frequency breakdown, we use only that part part of the diagram ou t lined by 0,0 2,0 2,2, because that area represents one icosa isce broken into two parts at th e edg e . VI me an s that the number of 'vertices in o ne icosa face of thi s par ticular br eakdown is 6. 2,2 e- 2,0 2- frequen cy Class I (" al ternate ") brea kdo wn FACE angle rtescr ibed by 1,1 0,0 1,0 E(L}: 9 means that t he number of edges in one icosa fa ce is 9. F(L} :4 means that ther e are 4 faces in one icos a face wh en broken into 2·frequency. The figures V(O) : 42, E(G) : 120, F(G) 80 ar e the number o f vertices, edges, and faces in an entire sphere based on th e 2-fr equency breakdown of the basi c icosa. • LENGTH 1,1 1,0 This means we are talking abo ut the st r ut that has vertex 1,1 at one end and vertex 1,0 at the other. This length number is also known as the CHORD FACTOR, and is called that in the rest of this book. To use it, multiply th is chord factor by the radius of the dome you want t o build. The result will be the length of that parti cular strut in the size dome you de sir e; it will be III th e same unit of measure as you measured the radius. In other words, if you give the radius in inches, the strut length will be qiven ill inches, This length is vertex to vertex. If you are using hubs, this numbe r will include the hubs, so the actual cut length of th« struts will be less than this number. It will be this number minus the diameter of a hub, It's easy to rnake a mistake here. Think it out carefully, and again in the morning, 1,1 st rut AX IAL ang le 1,0 I I I I '0.0 cen ter of sph ere DIHED RAL anqte is to tal incl uded an gle bet we en faces sharing 1,1 1,0 • AXIAL 0.0 1,0 1,1 refers to the angle between a line drawn from the center of the sphere (0.0) and vert ex LO, and the strut 1,0 1,1 (Fig. 2). It is the angle that a strut meets it s hu b from the side view. Note that it is not the entire included angle under a hub, but only t o t he line from t he hub to the sphere cent er. The value is DEGREES. 90° minus the axial angl e equals an gle you cu t or bend struts. • FACE 1,1 0,0 1,0 refers to one angl e of the triangle des cr ibed by these po ints . The anqle given wh ose apex is at th e second vertex identi fic at io n shown; in th is case, 0,0 (Fig. 3). Face angl es refer t o angles of tri angles generated by chord fac t or s, not spherical angles on a true sphere. Aga in, the number is given in degr ees. • 01 HEDRAL 1,1 1,0 refers to t he angle between the two faces th at share edge 1,1 1,0. It is t he total included angle and agai n is given in degr ees . See Fig. 4. The various " paragrap hs" in th e tabl es will yield all t he ne cessary informat io n if you keep in rnind that, just as the fac e in th e origi nal basi c icosahed ron was an equilateral triangle, t he 2-frequency fac e 0,0 2,0 2,2 is also equilateral and thus symmetric. Thi s means that th e three tri angle s shaded in Fig. 5 are exact ly the same. It also means that th e center tr iangle is equila ter al. 3-, 6·, 9-frequenc.ies (multiples of three ) have a point at the center o f the face inst ead of a triangle as in thi s case. Figs 6 and 7 show the 3 and 4 frequency breakdowns . Th e triangles of the same shading ar e t he same in all respects ex cept left and right or ientation. Note that the pattern of sameness is symmetric about the center of the triangular array of whatever breakdown frequency you are using. , 1 icosa face showong f-ACE ( 0( l. AXIAL I a: I, ,uld 01 HEDRAL (f3 ) angles. » >: 1\7' // V <\ I I-FIlEQl a ::'\c:Y ICOS,\ I1lWllO\ I I-I·HEVI F,NCr Ir:OS \IlI'.IJHON Ci ll•• I ;"i-I' HEOlJENU das" I (I. F HEl) lIE1\CY class 1 IIlf'l hod I IIu:ll.ml I II l1' thud 1 method I V(Ll '. 3. E( L) 3 F(Ll c 1 V( L) - 15 E(Ll - 30 F( Ll 16 V I L.) 21 E( L) " 45 FIll 25 V(L) 28 E( L) =63 F(L l 36 V (G) 12 E(G) - 30 F( G) = 20 V(G) - 162 E(G) 48 0 FIG) 320 V( G) ', 252 E(G) -r- 750 F(G) -r 500 V(G) , C 362 E(G) 1080 F(G) =720 LEN GTH 1,1 1.0 1.05 146272 LENGTH B 1. 1 1.0 0.29524181 LENGTH 1, 1 1.0 0.231 79025 A XIAL 0.0 1.0 1.1 58.282525 AXIAL D.O 1.0 1, 1 81. 510921 AX IAL 0.0 1.0 1. 1 83.34474 1 LtNGTH 1, 1 1.0 0.190476 86 FACE 1.1 0.0 1,0 59.999998 FACE 1,1 0.0 1.0 71. 33 1604 FACE 1. 1 0,0 1,0 71.5 90818 AXIAL 0.0 1.0 1.1 84. 534954 DIHEDRAL 1,1 1.0 138.189684 FACE 1,1 2. 1 1,0 60.159766 FACE 1. 1 2.1 1.0 61.79 7728 FACE 1,1 0,0 1.0 71.72 1\745 DI HEDRA L 1.1 1.0 172. 197790 DIHEDRAL 1.1 1,0 174. 186995 FACE 1.1 2,1 1,0 63. 14 1629 DIHEDRA L 1.1 1,0 175 .412319 LENGTH D 2.1 2.0 0.31286893 LENGTH 2.1 2.0 0.24724291 AXIAL 0.0 2,0 2, 1 80.D99996 AXIAL 0.0 2.0 2. 1 82.898845 LENGTH 2. 1 2.0 0.20281969 FACE 2, 1 1,0 2.0 63.668768 FACE 2. 1 1.0 2,0 65.444659 AXIAL 0.0 2.0 2,1 84.179635 FACE 2,1 3, 1 2,0 58.717473 FACE 2, 1 3, 1 2.0 59. 197807 FACE 2.1 1.0 2.0 66. 606874 DIHEDRA L 2.1 2,0 169.981901 DI HEDRAL 2,1 2,0 172.477804 FACE 2,1 3.1 2.0 60 .244049 DI HEDRAL 2.1 2,0 174 .131815 LENGTH C 3, 1 3,0 0.2945 3084 LENGTH 3. 1 3,0 0.24508578 AX IA L 0.0 3,0 3.1 81.5315 10 AX IA L 0.0 3,0 3, 1 82.96 1115 LEN GT H 3.1 3,0 0.20590774 FACE 3, 1 2.0 3,0 57.534353 FACE 3, 1 2,0 3,0 59 .964814 AXIAL 0.0 3, 0 3, 1 84 .090703 FA CE 3,1 4,1 3,0 59.920114 FACE 3.1 4.1 3,0 58.3693 23 FACE 3. 1 2,0 3,0 61.807500 DIHEDRAL 3. 1 3,0 169.617161 DIHE DRAL 3,1 3.0 171.544544 FACE 3, 1 4, 1 3.0 58.470501 2.FREQ UENCY ICOSAlIEIlI{():,\ class 1 DIHEDRAL 3, 1 3,0 173 .178778 met hod ) LENGTH E 3,2 3,1 0. 32491969 LENGTH 3,2 3,1 0.26 159810 V(Ll ,· 6 E(L) =9 F( Ll .. 4 AXIAL 0.0 3, 1 3,2 80.65029 2 AXIAL 0.0 3, 1 3,2 82.484 227 LENGTH 3,2 3,1 0.2 1535373 FACE 3,2 2, 1 3, 1 59.999 999 FACE 3,2 2,1 3.1 61. 671\566 A XIAL 0.0 3,1 3,2 83.818582 V( G) =42 E(GI 120 F(G) 80 FACE 3,2 4,2 3,1 62.565048 FACE 3,2 4. 2 3,1 59.99999 7 FACE 3,2 2,1 3,1 63.058991 DIHEDRAL 3.2 3, 1 169.642082 DIHE DRAL 3.2 3,1 171.553503 FACE 3,2 4.2 3, 1 59.611143 LENGTH A 1.1 1,0 0.61803 399 DIHEDRAL 3. 2 3.1 173.202995 AXI AL 0.0 1,0 1.1 71.999996 LEN GTH A 4. 1 4,0 0.25318459 LEN GTH 4,1 1\,0 0.22568578 FACE 1,1 0.0 1.0 68.86 1974 AXIAL 0.0 4,0 4, 1 82.727277 AXI AL 0.0 4,0 4,1 83.520774 LENGTH 4,1 4,0 0.19801258 FACE 1,1 2,1 1,0 59.999098 FACE 4,1 3,0 4,0 54.334194 FACE 4, 1 3.0 4,0 56.1 2501 7 AXIAL 0.0 4,0 4, 1 84 .31804 6 DIHEDRAL 1,1 1,0 161.9708 92 DIHE DRAL 4, 1 4,0 169.490046 FACE 4,1 5.1 4,0 59.1011 35 FACE 4,1 3,0 4,0 57. 948445 DI HEDRA L 4, 1 4,0 171.7301 90 FACE 4,1 5,1 4,0 57. 948447 LENGTH B 2, 1 2.0 0.54653306 LENGTH f 4.2 4,1 0.29858813 DIHEDRA L 4,1 4.0 172.856775 AXIAL 0.0 2,0 2, 1 74.141260 AX I A L 0.0 4,1 4.2 8 1.413979 LENGTH 4,2 4,1 0.25510701 FACE 2.1 1,0 2,0 55.5690 10 FACE 4. 2 3, 1 4. 1 58.796873 AXIAL 0.0 4,1 4,2 82 .670021 ILEN GTH 4,2 4, 1 0.216 62821 DIHEDRAL 2, 1 2,0 157.541075 DIHEDRA L 4. 7. 4,1 169. 505737 FACE 4, 2 3,1 4,1 59.16271 2 AXIAL 0.0 4,1 4. 2 83. "18 1857 FACE 4,2 5,2 4 1 62.43 2868 FACE 4,2 3, 1 4,1 60 .19441 DIHEDRAL 4,2 4,1 • , FACE 4.2 5,2 4,1 60. 1944'L7 171 .745039 0 1HEDRAL 4.2 4,1 172.843306 LENGTH 5. 1 5,0 0.1981471\3 AX IAL O.U 5,0 5, 1 84 .3141 62 LENGTH 5,1 5,0 0.18190825 FACE 5,1 4,0 5,0 54 .204537 AXI A L 0.0 5,0 5,1 84.78 1497 DIHEDRAL 5,1 5,0 171.765753 FACE 5, 1 4.0 5,0 55.403753 FACE 5,1 6.1 5,0 58.429183 LENGTH 5,2 5, 1 0.23'159760 I DI HEDRA L 5,1 5.0 173.170748 AX IAL 0.0 5,1 5,2 83.350295 FACE 5,2 4,1 5,1 58.430325 LENGTH 5, 2 5,1 0.20590773 2·frcquency . . f f DIHEDRAL 5.2 5,1 171.7839 24 AXIAL 0.0 5,1 5, 2 84.090701 sh"de<1 faces Fi r:. ? Fac('!s t he same FACE 5.2 4.1 5,1 58.470502 desiq ncd - I",,"""'",,' . co lor are ident ical LENGTH 5,3 5.2 0.24534642 FACE 5,2 6. 2 5, 1 61 .807496 ( ears ago. except that white AXIAL 0.0 5.2 5.3 82.9535 90 DIHEDRAL 5,2 5.1 173.178770 f aces occu r i n led , and pHi r s as " left s" FACE 5,3 4,2 5. 2 60.070368 ornes. due and "ri gh t s," DI HEDRA L 5.3 5,2 171.838518 ILENGTH 5,3 5, 2 0.21535373 i lk. Bc for e f'i,!. 7 AX IAL 0.0 5.2 5,3 83.818583 --- - - FACE 5, 3 4,2 5,2 59.611144 I I FACE 5,3 6.3 5,2 63.058989 and I IDIHEDRAL 5,3 5,2 173 .202989 fe rent . H ... - \ \ ( £1 . /\ ,-- 1\ ' , ",. \ LENGTH 6,1 6,0 0.16256722 A XIAL 0.0 6,0 6,1 85.337645 FACE 6, 1 5,0 6,0 54.137622 DIHEDRAL 6,1 6.0 173.240530 .-. :-'.J LENGTH 6,2 6.1 0.18738340 .....", liNl,M' ; :1. .. IAXIAL 0.0 6,1 6,2 84.623971 A B A •• ICOSAIIEDHOi\ 1 \ /\ 7\ 7 iIifI!" I ••• • FACE 6,2 5.1 6,1 57.989368 nu,thorf t .. DIHEDRAL 6, 2 6. 1 173.254913 V(L) 10 ElL) ·, 18 F{l) · 9 '\( * 'Y' V(G l 92 E( G) 270 F(G) O' 180 LENGTH 6,3 6,2 0.20 28 1970 AXIAL 0.0 6,2 6,3 84 .179636 LENGTH 8 1,1 1,0 0.40354821 L C FACE 6,3 5,2 6, 2 60.244049 AX IA L 0.0 1.0 1,1 78.359272 DIHEDRAL 6,3 6,2 173.318399 FACE 1.1 0,0 1.0 70.730537 FACE 1,1 2.1 1.0 58.583164 DIHEDRAL 1,1 1,0 168. 641064 \ I \ I I aLL ...-1. 0 11 pagf! an' LEN GTH C 2. 1 2.0 0.4124 1149 \ / I " , ... II Churd fur: A XIAL 0.0 2, 0 2,1 78.099906 .. j ..,'!t- . , ) [Al ternato) FACE 2, 1 1,0 2,0 60.7084 16 3·f rL'queflcv. method) FA CE 2.1 3,1 2,0 60.708416 f' ace s Ihe s", ne DIHEDRAL 2. 1 2.0 166.42 1442 col or cue identi cal LENG TH A 3,1 3,0 0.34861548 Fi/!. (j AXIAL 0.0 3.0 3.1 79.961 621 FACE 3,1 2, 0 3.0 54.634727 DIHEDRAL 3.1 3,0 165.564739 LENGTH 8 3,2 3,1 0.40354822 AXIAL 0.0 3,1 3,2 78.359272 FACE 3,2 2.1 3,1 58.5831 64 I .. .. ... .; -"- .L .. urm mT0I151 emRrH;1.E5i DIHEDRAL 3,2 3,1 165.542280 .- -- con/tnt.tU frtJl1/ LENGTH-lO 3,2 2,3 .333333 AXI AL 2, 3 82, 5461 AX IAL 3,2 82. 5461 LENGTH-l 1 3,2 3,3 .245992 AXIAL 3,3 81.6616 AXIAL 3,2 82.5381 LEN GTH·12 3,2 4,2 .247825 AX IAL 4,2 78.1855 AXI AL 3,2 '78 AX IAL 3,2 79.951 1 TR IANGL E 2 LENGT H-13 4,1 3, 1 .246953 AXIAL 3, 1 74 .9J2"7 AXIAL 4,1 77. 1686 LENGTH ·14 3, 1 2,1 .224244 ;\ XI AL. 2,1 73.3502 AXIA L 3, 1 73.409 1 LENGTH·1 5 2, 1 1,1 .190"703 AXIAL 1,1 "7 9. 2185 AX IA L 2,1 77.8740 LENGTH · 16 4.1 3,2 .329 151 AX IAL 3,2 79.8255 AX IAL 4, 1 80. 7485 LEN GTH 17 3,1 3,2 .33949 1 AX IAL 3,2 79 ."1994 AX IA L 3,1 79. 1286 LENGTH -18 3,1 2,2 _331518 AX IA L 2,2 "18.5470 AX IA L 3,1 78.8638 LEN GTH-19 2, 1 2.2 .329151 AXIAL 2,2 78.8481 AXIA L 2.1 79. 1186 LENGTH-20 2.1 1..2 .303528 AX IAL 1.2 80.3767 AXI AL 2.1 79, 7090 LENGTH-21 3.2 2,3 .303528 AXIAL 2,3 79,7090 AXIA L 3,2 80.3767 LENGT H-23 3,2 2,2 .226708 AXIAL 2, 2 73.691 b A XI AL 3, 2 75. 1394 LENGTH-22 3, 2 3,3 .276229 AXIAL 3,3 82.5798 A XIAL 3,2 82.2006 The foll owing compar ison between Class I (Alternat e] and Cl ass rI (Tri acon) breakdown is r epr int ed fr om Edward Popko' s Geodesics. THE TRIACON BREAKDOWN The triacon br eak down has those advantaaes over the ot he r breakdowns a. A m inimum number of d iff eren t co mpo nents h. A symmetry of re lat ions hip 01' adjacent f aces ma king eas y co mb inat io n in to diamonds. an d t he follo wing di sadvant ages : a. A great er var iati o n o f mem ber len gth th an with th e Alternat e sys te m. b. No co mp lete gr ea t ci rcle delineat ed na t urally by t he s truc t ura l paner n-vsu ch as t he equatori al ob ta ina ble o n th e eve n nu mber Alte rna te breakdown s. Co nse quent ly t runcate d base member s must be used in every case . c, r-rp. quende.smus t al ways run in <I n even numbe r..-t herct ore t here is less graduation in sca le th an is availa ble wi th the Alt ern at e breakdown . In pr act ice, th e adva nta ges of th e Tri aco n become more e mpha tic in th e highe r f ruqu en cie s-e-usuat tv th is mean s in th e larger diam et er st ruc t ur es such as th ose 100 feet Or more.. . THE ALTERNATE 8RE AKDOWN The alternate brea kdown has th e fo llo wi ng adv an tages : a. A m inimu m vari ati on in member le ngt h, and excp.pt for the pe nt joi nt s, less var iati on In face an gles. b. In even freq uen cies, a cont inuous equ at or is delinea ted , so ach ieving hemispher es wit ho ut t he need f or trun cat ed members at th e base. c. As odd and even number frequ e nci es a re bot h both obtainable, a more gradu al variat ion in sca le o f br ea kdown is avai lab le. Di saovantaqes : a. Th e number of different co mponen ts in re lation to fr!::' qu e ncy incr ease on a geome tric scal e... 0, 0 1,1 2,1 1,0 2,1 3,1 3,0 3,1 2,1 3,2 1,0 1,1 2,1 1,0 2,0 2,0 2,1 3, 1 2,0 3, 1 3,0 3, 0 3,1 3,1 3,2 1,0 1,0 3,1 1,0 3,1 2,0 2,0 2,0 3,0 3,0 1,0 1,0 1,0 2,0 2,0 2,0 3,0 3,0 3, 1 3, 1 0,0 1,0 1,0 2, 1 2,1 1,0 2,0 2,0 3, 1 2,1 3,1 2,0 3,0 2,1 3, 1 1,0 1,0 1,0 2,0 2,0 2,0 3,0 3,0 3, 1 3,1 0,0 2,1 2, 1 1,1 3,2 1,0 3, 1 2,1 2,0 3,1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1,1 0,0 1,1 1,1 1,0 2,1 1,0 2,1 2,1 2,0 2,0 3,1 2,0 3,2 2,1 0, 0 1, i 2, 1 1.0 2,1 3,1 2,0 3,1 2,1 3, 2 LENGTH-A LENGTH-A LENGTH-A LENGTH·B LENGTH-B LENGTH-C LENGTH-C LENGTH·D LENGTH-E LENGTH-F AXIAL AXIAL AXIAL A XIAL AXIAL AXIAL AXIAL AXIAL AXIAL AXIAL FACE FACE FA CE FACE FA CE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE DIH EDRAL DIHEDRAL DIHEDRAL DI HEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DI HEDRAL 0,0 1,1 6·FREQUENCY ICOSAHEDRON class]] method 3 .22425 .22425 .22425 .26427 .26427 .21877 .21877 .25562 .20604 .231H2 83.50440 82.40690 83.5044 0 83.72004 82.65686 83.72004 84.08695 83 .31\394 83.504 34 82.40696 71.46610 54.26700 54.26 700 71.46610 53. 58140 70.090 20 56.3 2840 56.32840 70.09020 53.58140 54.45830 66.0 9890 57.27480 71.46610 54.26700 170.60280 174.87730 170.90110 170.68770 174.65000 171. 64300 171.47280 174.06170 171. 64300 174.87730 3,0 !J.FREQUENCY ICOSAHEn RON II . mclbtll1 3 LENGTH 0,0 ,0 0.1702t"i 388 L. ENGTH 1,0 1,1 0.19937 078 LENGTH 1,0 2,0 0 .167 016 11\ LENGTH 2, 0 7, 1 0.1%(-)3944 L [ NGTH 2,0 3, 0 0.16103 172 LENGT H 3,0 3,1 0 .181\9354'1 LENGTH 3,0 4,0 0 .155331\60 LENGTH 4.0 4.1 0. 1687 1032 AXIAl. 0.0 0,0 1,0 85 0 7' 0" A XIAL 0.0 1,0 1,0 84° 16' 44" AX IA L 0.0 1.0 2,0 85 0 12' 36" AXIAL 0.0 2,0 1.1 84°23'1 1" AXI AL 0.0 2,0 3,0 85° 22' 58" AXIAL 0.0 3,0 2, 1 84°41' 40" AXIAL 0.0 3,0 4,0 85° 36' 11" AXIAL 0.0 4,0 3. 1 85° 9' 41" We don't have face and di hedral angles f or 8 fr equency . I}I di agr am abo ve s ho ws one of th e 0 ident ica l right tri angl es in th e icosa face. l cose fa ce shown by dotted lines 1),0 I I I I , I 'Vf-- .> 4/1 I \ I \ , I \ I \ I \ I \ 4,'2... I \ I \ I :2 4 i O 4·FREQUENCY ICOSAHEDRON class II method 3 LENGTH·A 0,0 1,0 .33609 LENGTH·B 1,1 1,0 .38948 LENGTH-A 2,1 1,0 .33609 LENGTH-C 1,0 2,0 .31337 LENGTH-D 2, 1 2,0 .36284 A XIAL 0.0 1,0 0,0 80. 31740 AXI AL 0.0 1,0 1,1 78.77250 AXI AL 0.0 1,0 2,1 80.31740 AX IA L 0.0 2,0 1,0 80. 98560 AXI AL 0.0 2,0 2, 1 79. 54750 FACE 1,1 U,U 1,0 70.81977 FACE 0,0 1,0 1,1 54.58971 FA CE 1,1 1,0 2,1 54.58971 FACE 1.1 2, 1 1,0 70.81977 FACE 1,0 2,1 2,0 53.10 875 FACE 2,1 1,0 2,0 67.82582 FACE 1,0 2,0 2,1 59.06569 DIHEDRAL 0,0 1,0 166.04826 !HEDRAL 1,1 1,0 172.07843 DI HEDRA L 2,1 1,0 167.17545 orl-l EDRA L '1, 0 2,0 166.30846 D! H'EDRA L 2, 1 2,0 171.37576 2,0 3/2 1,1 1, 0 1,0 0640851 1,0 0.713644 1,0 0,0 71.31131 1,0 1,1 69.09484 0,0 1,0 67.66866 1,0 1,1 56.16566 1,0 153.78959 1,0 161.94600 Below are chord fact or s f or Class II , Meth od 3, what we call "tri acon" in the rest of th e book. The f igures are based 011 Class II geodesic spheres as developed by R. Buckminster Fuller. The numbers are from the computer readout generated by program s developed by Joseph D. CIinton under a NASA-sponsored research grant. " Advanced Structural Design Concepts for Future Space Missions," Final Report, March, 1970 NASA Contract NGR 14-008-002 The general instructions 011 p. 108 app ly t o the below chord factors end ,mgles, and to all the chord factors and angles ill the book: The tabl es f or Cl ass II fol l ow the same f or mat as tho sef or Cl ass I except th at the verti ces are ident ifie d as in th e diagram: On Ihispage aro Ch<ml Flid ortl for: ..flAM II [Tri acou) method 3 LENGTH-A 0,0 LENGTH B 1.1 AXIAL 0.0 AXIAL 0.0 FACE 1,1 FACE 0,0 DIHEDRAL 0,0 DIH EDRAL 1,1 2-Io'REQUENCY ICOSAHEDRON class II method 'I I bY vertex ide nt if icati on di agram for Class II (" tri aeon" l brea kdown. From do tted line up is one of th e 6 ident ical right tri angl es in the icosa face. St ruts shown in thi s right t riangle are re pate d in t he ot he r five right tr lanpt es of th e l ace ; see di agram right . I I i III .5627 12 83.2072 82 .6188 .58 7072 83.4997 83.4904 .449176 84. 3987 84 .7067 .420724 80.9436 80,4348 .4 17835 80.5673 BO.8345 .434271 81.2813 81 .1842 .420724 8 1.2969 81 .2 125 .400996 81.7135 81.9592 .400996 81.9592 81.7 13 5 .311805 81.9171 82 . 1298 .57 1025 83 .5990 83.340 1 ZAF U .284 682 84 .8883 84 .7560 .339072 82 ,9303 82. 1528 .294883 80.9677 79.3063 .333333 83 .9348 83.9348 .34651 3 82.8373 83.3 116 .360 162 82.8 744 82.6149 .340 16 1 81 .3366 82 . 178 1 .34 6513 81 .7785 81,2158 .3 13561 82.0800 80.9075 .333333 75.305 1 75.3051 .382492 77.0644 75.3257 .369767 76.1465 76.8214 .41 6405 79.4932 78. 31 96 .38249 2 78.9312 .450817 79.0514 80 .6253 .333333 78.9 158 78.9 '158 .372 230 83.2 359 82 .6929 ,4911 14 832419 82 .5023 On this pag" arc Chord for: Elliplieal Domes TRIACON BREAKDOWN FREOUENCY - 4 EXPANSION ' 0.61800 TRIANGLE 1 L ENGTH -l 2, 1 1, 1 AXI A L 1,1 AXI A L 2,1 LENGTH-2 3, 1 2.1 AXIAL 2.1 AXIAL 3, 1 LENGTH-3 4,1 3, 1 AXIAL 3, 1 AXI A L 4, 1 LENGTH-4 2.1 1,2 A XI A L 1,2 AXIAL 2, 1 LENGTH· 5 2,1 2,2 AXIAL 2,2 AXI A L 2, 1 L ENGTH-6 3,1 2,2 A XI A L 2,2 A XI A L 3,1 L ENGT H· 7 3.1 3,2 AXIAL 3,2 A X I A L 3,1 LENGTH - 8 4,1 3,2 AXIAL 3,2 AXIAL 4,1 LENGTH -9 3,2 2,2 AX IAL 2,2 AXIAL 3,2 CP11.t!nt/e.:! OfJ FRcrOA5I Drd EGG ZAFU .620184 78.1926 76 .534 9 .7 1364 3 75.4669 75.4669 .57972 3 72.0687 75.5755 .53 0577 62.171 3 70.2740 .69 2035 61 .3078 67 6662 . 713643 62.8087 62.8087 .7784 58 73.6339 70.2562 1. 05144 75. 8262 71.9315 EGG .607478 72.9138 708042 .402019 63.6428 60 .2092 .649834 67. 5946 67 .5946 .29 1222 76.8189 77.6136 .418726 76.1561 78.3853 .498661 80.2670 81.6644 2. 2 .720932 72.286 5 73.2628 1, 1 1.03086 77.714 5 78.3811 2.1 1. 1 1.03086 1, 1 77.714 5 2,1 78.3811 2, 1 1,2 .858506 1,2 72.5488 2,1 72.5488 TRI ACON BREAKDOWN FREOUENCY = 2 EXPANSION = 1.6 1800 TR IANGLE 1 LENGT H-A 2, 1 1,1 AXIAL 1,1 AXIAL 2,1 L ENGTH - B 2.1 1,2 A XIAL 1,2 A XI A L 2, 1 L ENGTH -C 2,1 2,2 A XI A L 2,2 A XI A L 2.1 LENGTH · D 2.1 3.1 A XI A L 3.1 AXIA L 2, 1 T RIANGLE 2 LENGTH-E 2. 1 AXI A L 2,2 AXIAL 2,1 L ENGT H- F 2, 1 AXI A L 1,1 AXIAL 2, 1 LENGTH-F A XIAL A XIAL LENGTH-G AXIAL AXIAL TRI ACON BREAKDOWN FREOUENC" EXP ANSION · 0.61800 T RI A NGI, l I LENGTH -A 2.1 1, 1 A XIA L 1, 1 AXI A L 2,1 LENGT H-S 2,1 1,2 AXIAL 1,2 A XI A L 2, 1 L EI\IGT H· C 2,1 2,2 AXI A L 2,2 AXIAL 2.1 LENGT H-D 2, 13,1 A XI A L 3, 1 AXIAL 2.1 TRIAI\IGLE 2 LENGTH -E 2,1 2,2 A XI A L 2,2 A XI A L 2,1 LEN GTH-F 2,1 1,1 AXIAL 1,1 AXIAL 2, 1 LENGTH·G 2, 1 1,2 A XIAL 1,2 AXIAL 2,1 TR IACON BREAKDOWN FREOUENCY 4 EXPANSION 1.61800 TR IANGLE 1 L ENGT H- l 2,1 1,1 A XIAL 1, 1 AXIAL 2,1 LE NGTH·2 3, 1 2.1 AXI A L 2, 1 A X IAL 3, 1 L ENGT H-3 4, 1 3,1 AXIAL 3,1 AXIAL 4,1 .345326 83.7568 83 51 52 .3674 G5 81 8237 806379 2704 13 80 .5045 78.7365 .403548 82 .5592 82.5592 .396438 82 .0864 81 .3232 .408974 81.2193 8 1. 5911 .348879 80.1293 782346 .403547 80.2827 80.2827 . 367465 80 .6379 79. 3 178 .344 527 81 .1816 81.5156 .322275 75.1 273 77 .2476 .300028 74.6243 76.8781 .27312 1 77.4455 78.849 7 .4 12220 78.3722 78.4240 .300536 73.95 13 74.0238 .300028 74. 624 3 74. 624 3 ,403548 78.6308 78.63 08 fig. 2 1, 0 1.0 2.0 2,1 2.0 ioos» edge f::,Z 1,0 1,0 .403548 1,0 78.2503 1.1 78.250 3 line o f sy mmetr y TRIANGLE 2 L ENGTH· l 1 3,1 AXI A L 2, 1 AXIAL 3, 1 LENGT H· 12 3, 1 AXIAL 2,0 AXI A L 3, 1 LENGTH-13 3, 0 2.0 AXIAL 2,0 AXIAL 3,0 LENGTH·14 2,1 AXIAL 2,0 AX I AL 2,1 L ENGT H- 15 2, 1 A XI A L 1,0 AXIAL 2.1 LE I\IGTH· 16 2.0 1,0 A XI AL 1.0 A XI A L 2,0 L ENGTH · 17 1, 1 A XIAL 1.0 AXIAL 1. 1 1S t .' . strut length labels and vertex tebols for a 4v Triscon. f ig. 3 ZA FU TRIACON ELLIPTICAL This diagram [fig. 2) dem onstra tes th e co ngr uent t riangles an d str u t s In a 2v. Th e same sy mmetri es hold f o r the 4v With the addi t ional fact that fo r lh e t rranqlc 1' s t he edqe of the ico sa (dotte d hnes) i s al so a l ine of symmetry and th e , ; rlJlJ repeated on both sides of i t w ill bl t he same. Th e ven: ic'-)s In thf! 2 baSIC lr ii:lngles l abeled 1,1:1' t his f or t ri acon : .s LENGTH- 17 A XI A L AXI AL For the st ru t l ength l abel f or 2v see fig . 12) and fo r ver tex label s use up ' to 2,2 on f ig. (3), TherR i s a paper I I model of the 'Lv ex pansion 0.618 on " ·' 24 . .6 18033 65,4414 65.4414 .74 3487 69.7882 73.6147 .749 689 74.4582 77 7135 55 5583 73 .3 128 72.8299 .618033 72.0000 72.0000 .9 105'16 77 81129 76.6834 .788849 79 .0865 78. 3322 .788849 78.332 2 79.0865 .357081 J 74.0175 75. 3841 .485463 75.5989 78.2050 .498249 8 1.4523 82 .5144 .4 03548 72.5908 72.5908 .45 1555 73.7840 76.0435 .42 1274 75.7580 75.0177 .544692 78.4663 80 .1690 77. 2521 77,252 1 .48546 3 78.2050 79.4305 . 359094 79.5859 79. 3353 .586029 81.563 1 80 .8686 .594565 82.0554 81.3864 .494368 82.4553 82.8 739 ,494368 82.8739 82.4553 .4129 10 78 .0031 77 .98 11 .6155 26 81.984 7 81.9701 .594565 82.0554 82.0554 1,0 0.0 2,0 2,0 1,0 1,0 2,0 2,1 3, 1 3,0 1,0 1,0 2,0 1,0 1,1 1,0 1. 1 1, 0 2,1 2,1 2. 1 2,0 2,1 al aO diar[:3-V 3,3( y;>.... t A V:7r' " LENGTH-8 1,1 1,0 A XIAL 1, 0 AXIAL 1, 1 LENGTH·C 2, 1 1.0 AXI A L 1,0 AXIAL 2,1 LE NGTH- O 2,0 1,0 AXI AL 1, 0 AXI .!\ L 2,0 LENGTH: E 2,1 2,0 AX I A L. 2,0 AXIAL 2,1 T"l II'NGL E 2 LEN(Jn l. 13 1. 1 1,0 A X I A L 1,0 .:\XI A L 1. 1 L!: NGTH. F 2,1 1,0 AXIAL 1, 0 A X I A L 2, 1 LEN GTH- G 2,0 1.0 A X I A L 1,0 A X I A L 2,0 L EN GT H-G 1,0 0,0 A XI AL 0.0 AX IA l 1,0 A LTERNATE 8RE AKDOWN FREOUEN CY = 3 EGG EXPANSION = 1.61800 TRIANGLE 1 "L ENGT H- l AX IAL AXIAL LENGTH·2 A XI A L AX IAL LENGTH-3 A XIA L AXIAL LENGT H-4 AXIAL AXIAL LENGTH -5 A XI A L A XI A L L ENGT H-6 A XIAL AXI A L L ENGT H- 7 A XI A L A XI A L , LENGTH· 8 AXI AL AXIAL LE NGTH-9 ,1, XI A L AX IAL l ENGT H- lO AXIAL AX IAL EGG ,618033 72 0000 72 .0000 .459493 65.4404 69.788 1 .41 8429 69.0883 71 .9665 .418429 71.9265 69.0883 .53377 1 80.0425 79.0533 .618033 77. 893 7 77.8937 ,56277 3 76.6841 73.6152 .445083 76.8362 7:l ,0428 .543 03 5 7G. 1696 76. 7847 .578607 65 .314 6 69.6860 dl8q!a m 1br2y 2,0 1,0 1,0 0,0 1.0 1,0 0,0 1,0 1, 1 1.0 1,1 2,1 1,0 2.1 2.0 1,0 2,0 2,1 2,1 2. 1 Z,o{ '" l--... ,'" )Z,o expensi on ,.efe" t o th e 3"10unt of drs to r uon t he zenit b-ro..nJ'I,t (an ex pans ion ,;, 1 b0i ng a rLIh"' ''' t hu"' 1 str et chedno rn e: an less tha n 1 JlJuashed }. K'n,p i n mi " llt hal the chord s 10 LIJ le l t 01 ' hE! l in' 0 1sv rn rne tr v, ftp. I,l l , }w t h- sarns .' '; ones t o Ul€ rlg,tlT . 11d th ai the 2 1) I>SIl m angles sha. /l carrai n chords i n II>!! [est row. Lenot h d3111 con be ill I he vvov <'\ fW rJ. 101 th e ax i al anul!'; may be diii Hr . 11I 1l1 " ach end of the stru; so t he coo rdinate If IJITie s. whi ch end of t he st r ut It ld moll! ref er s to . 1n co nst r uc u on 8'.Qgl - .ngles con. be rounded off a [Jj;Jtfse Of two (o r convc ruence . I am nl U pu bli shi ng dihedral auqles they are rarel y used (ex cept In sun d ome t ype con structi on) and wo uld take up a lot of space. T he 'aco anoles call be cal cul ated uSi ng t he law o f cosmej . If yo u need the d lhedrel anqles, so me spec ral ex pansi o n, o r truouencv , w rit e and I ' ll send th em lor computer cast s. Ther e are 2 dif l erent basic u i " ngl el i n th e el l i pt ical Icosahedron and anv br eak down Wi ll re sult rn a drf h 'Tt' ., t st ru t Ie rrqt h i n th ese two triangles. l' he [" lJles con rc .n i nformati on '.in 1/ .l d l of tho se t ri " ' l ylt'3. ... AI . ' /f\,, {J--c, 'Z. TRIANGLE 2 LE NGTH - 8 1, 1 1,0 AXIA L 1,0 AXIAL 1, 1 LENGTH-F 2, 1 1,0 AX IAL 1,0 AXIAL 2.1 LENGTH-G 2,0 1,0 AXIAL 1,0 AXIAL 2,0 LE NGTH-G 1,0 0.0 AXIAL 0,0 A XI A L 1.0 A LTERNATE BREA KDOWN FREOUENCY 2 ZAFU EX PANSION = 0.61800 TRI ANGLE 1 LENGTH- A AXIAL A XI AL LENGTH -B AXIAL AXI AL LENGT H-C AXIAL AXIAL LENGTH· D A XI A L A XI A L LENGTH-E AXIAL AXIAL ALTERNATE ELLIPTICAL The vert ices arc l abeled the same W<lY as with the spber lcal, startmg With 0,0 at each end of the t wo bas ic t ri angles. .0,0 -.. D AI ;! "i;, - "' B :J o & c ." '; AL TERNATE3REAKDOWN FFtE OUENC Y 2 E>:I' A NSI ON . 1.61800 I I A NGL E I L ENGT H-A 1,0 0,0 A XIAL 0, 0 AXIAL 1,0 TRIANG LE 2 I I LENGTH- l 1 3,1 2.1 AXIAL 2, 1 AX IAL 3,1 LENGT H-12 3, 1 2,0 AXIAL 2,0 AXI AL 3, 1 LE NGTH·13 1,0 0,0 AXIAL 0,0 t\XllI L 1,0 L ENGTH-13 3,0 2.0 A X I A L 2,0 AX IAL 3,0 LENGTH· 14 2,1 2,0 AXIAL 2,0 A X I A L 2,1 LENGT H· 15 2,1 1,1 AX IAL 1.0 /, XI A L 2, 1 L ENGT H· 16 2,0 1,0 A XI A L 1,0 A XI A L 2,0 lati vel y tool s, or afte r ; were J f laqe is me " barren. ut refu l Jring .ment bette r one LJ sed Jmes. .linqs. avis ivls. le y ou f ir II anvo ns lon e. Jod rt h I l ot ,nizat ion. 3m st 40 new , as one t s, are e deast LJ sed ses Figures given below are for Class I (AI crna I, Figures below are for CI I I ITri can). Figures below are for domes gen re l rorn Tlbl,- .. ' t l Method 2. Edges of the icosa face ar divided Method 1. This methoc will pro bly the tetrahedron and octahedron. They are Cut struts il into equal parts, the tr iangles are more produce smoother arcs than the triacon not as round as domes generated from the get a metri c equilateral, the dome is structurally stronger, chord factors we have used, as shown on icosahedron. The octahedron forms a but you do not get the smooth arcs of p. 110 . We have not built any domes natural t runcation at the hemisphere. uyna Strut N. Method 1. Make models of a given frequency with this method. Domes uses a 4-frequency octahedron in both methods to see the differences. We breakdown for th eir domes, For assembly have not built any domes with this method. diagram see vertex identification diagram, A 3C p.108, 30 B1 30 C 75 75 82 75 CLASS I CL ASS II 4-FREOUENCY OCTAHEDRON, CLASS I 8-F REQUENCY TETR AHEDRON Second stet. METHOD 2 METHOD 1 METHOD 1 LENGTH 1,1 1,0 .2425356 Important: LENGTH-B 1,1 1,0 0.42406?-5 LENGTH·A 0,0 1.0 .3091073 LENGTH-S 1,1 1,0 .4472135 L.ENGTH 2,1 2,0 .2917975 lA's and C', LENGTH-C 2,1 2,0 0.4041944 LENGTH -G 2,1 1,0 ,3466883 LENGTH-D 2, 1 2,0 .5 176380 LENGTH 3,1 3,0 .3319301 to len gth, Y LENGTH-A 3, 1 3,0 0. 3669588 LENGTH-E 1,0 2,0 .34034 24 LENGTH-C 3, 1 3,0 .4388710 LEN GTH 3,2 3,1 .3779644 LEN GTH -A 3,2 3, 1 0.3669588 LENGTH-D 2, 1 2,0 .36284 33 LENGT H· E 3,2 3,1 .5773502 LEN GTH 4,1 4,0 .3203644 Start with s AXIAL 0.0 1,0 1,1 77.75857 LENGTH-C 1,1 1,0 ,3590112 I.ENGTH-A 4,1 4,0 .3203644 LENGTH 4,2 4,1 .4595058 AXIAL 0.0 2,0 2,1 78.34037 AXIAL 0.0 , 1,0 0,0 81.10908 LENGTH-F 4,2 4,1 .4595058 LENGTH 5,1 5,0 .2491473 tha n final Ie AXIAL 0.0 3,0 3, 1 79.42750 AXIA L 0.0 1,0 1,1 79. 65903 AXIAL 0.0 1,0 1,1 77.07903 LEN GTH 5,2 5,1 .4388710 margin if yc AXIAL 0.0 3,1 3, 2 79.4 2150 AXI AL 0.0 1,0 2, 1 80.01768 AXIAL 0.0 2,0 2, 1 74.99999 LENGTH 5,3 5,2 .5773502 FACE 1,1 0,0 1,0 70.59285 AXI AL 0.0 2,0 1,0 80.20222 AXIAL 0.0 3,0 3,1 77.32411 LENGTH 6,1 6,0 .1898393 Once the ar FACE 1,1 2,1 1,0 63.27959 AXIAL 0,0 2,0 2,1 79.54741 AXIAL 0.0 3,1 3,2 73,22134 LENGTH 6,2 6,1 .3 319301 Cut enough FACE 2, 1 1,0 2,0 63.00334 FACE 1,1 0,0 1,0 71.00277 AXIAL 0,0 4,0 4,1 80.7B252 LENGTH 6,3 6,2 .51/6380 FAC E 2, 1 3, 1 2,0 58. 36019 FACE 0,0 '1,0 1,1 54.49861 A XI A L 0.0 4,1 4,2 76.71747 LEN GTH 7, 1 7,0 . 1598888 FACE 3, 1 2,0 3,0 54.70 356 FACE 1,1 1,0 2,1 58.81706 FACE 1,1 0,0 1,0 88.52971 LEN GTH 7,2 7,1 .2503401 A rad ial an FACE 3,2 2,1 3,1 53.99 331 !"ACE 1,1 2,1 1,0 62.36587 FACE 1.1 2,1 1,0 61.26167 LENGTH 7,3 7,2 .37504 75 to th e same DIHEDRAL 1,1 1,0 169. 34601 FACE 1,0 2,1 2,0 57. 27357 FACE 2,1 1,0 2,0 70.3 2347 LENGTH 7,4 7,3 .4472136 crro: in st ru DIHEDRAL 2,1 2,0 166.U6061 FACE 2,1 1,0 2,0 63.75086 FACE 2,1 3,1 2, 0 56.10466 LENGTH 8,1 8,0 . 1403741 agains t a rna DIHEDRAL 3,1 3, 0 164.81383 FACE 1,0 2,0 2, 1 58.97555 FACE 3, 1 2,0 3, 0 52.97068 LENGTH 8,2 8,1 .1990172 DIHEDRAL 3,2 3,1 164. 40862 DIHEDRAL 0,0 1,0 167.1 86 2 FACE 3,1 4,1 3,0 59. 36916 ENGTH 8,3 8,2 .2747709 If you can't DIHEDRAL 1,1 1,0 170.9164 FACE 3,2 2,1 3, 1 59.99999 LENGTH 8,4 8,3 . 3382039 for t his type DIHEDRAL 2. 1 1,0 167. 8084 FACE 3,2 4,2 3,1 67.79066 AXIAL 0.0 1,0 1,1 83.03471 DIHEDRAL 1,0 2,0 167.2566 FACE 4,1 3,0 4,0 45.7351 3 AXIAL 0.0 2,0 2,1 81.61066 01HEDRAL 2,1 2,0 169.5607 FACE 4,2 3,1 4,1 56.70583 AXIAL 0.0 3,0 3, 1 80.44669 DIHEDRAL 1,1 1,0 172.4355 AXIAL 0.0 3, 1 3,2 79. 10660 DIHEDRAL 2, 1 2,0 164.1306 AXIAL 0.0 4,0 4, 1 80.78252 DIHEDRAL 3,1 3,0 162.5781 AXIAL 0.0 4,1 4, 2 76. 71747 DIHEDRAL 3,2 3,1 162.9038 AXIAL 0.0 5,0 5,1 82.84385 DIHEDRAL 4,1 4,0 161.7990 AXI A L 0.0 5,1 5,2 77.32411 DIHEDRAL 4,2 4.1 162.1625 AXIAL 0.0 5,2 5.3 73. 22134 AXIAL 0.0 6,0 6,1 84.55 329 AXIAL 00 6,1 6,2 80.44669 ,----- AXIAL 0.0 6,2 6,3 74.99999 AXIAL 0.0 7,0 7,1 85.41462 AXIAL 0.0 7,1 7,2 82,80941 AXIAL 0.0 7.2 7,3 79.19168 AXIAL 0.0 7,3 7,4 77.07903 AXIAL 0,0 8,0 8,1 85.97526 AXIAL 0.0 8,1 8, 2 84.28912 AXIAL 0.0 8,2 8,3 82.10341 AXIAL 0.0 8,3 8,4 80.26438 FACE 1,1 0,0 1,0 119.5123 FACE 1,1 2,1 1,0 98.65558 -" FACE 2,1 1,0 2,0 108.2910 CLASS I &FREOUENCY1COSAHEDRON CLASS II 6-FREQUENCY OCTAHEDRON CLASS I FACE 2,1 3,1 2,0 81,79592 METHOD 2 METHOD 1 METHOD 1 FACE 3,1 2,0 3,0 89.25 280 LENGTH-E 1,1 1,0 ,3212440 LENGTH-A 0,0 1,0 .1999068 LENGTH 1,1 1,0 .2773501 FACE 3,1 4,1 3,0 60.31269 LENGTH-C 2,1 2,0 .3132066 ,LENGTH- S 1,1 1.0 .2338277 LENGTH 2,1 2,0 .3203644 FACE 3,2 2, 1 3,1 98.03357 LENGTH-D 3,1 3,0 .2977251 LENGTH· C 2,1 1,0 .2204 259 LENGTH 3, 1 3,0 .3319301 FACE 4;1 3,0 4,0 64.02030 LENGTH·S 3,2 3,1 .2995157 LENGTH-G 1,0 2,0 .2236135 LENGTH 3,2 3, 1 .3779644 FACE 4,1 5,1 4,0 45.73513 LENGTH-A 1,2 4,1 .2759:J44 LENGTH-F 2,1 2,0 .2438120 LENGTH 4,1 4,0 .2960307 FACE 4,2 3,1 4,1 80.82667 AXIAL 0.0 1" ,0 1,1 80.75699 LENGTH-H 3,1 2,0 .23119 14 LENGTH 4,2 4,1 . 3851750 FACE 3,2 4,2 3,1 69.40889 AXIAL 0.0 2,0 2,1 80.99019 LENGTH-J 2,0 3,0 .22 75411 LENGTH 5,1 5,0 .2419696 FACE 4,2 5,2 4,1 56.70583 Sh AXIAL 0.0 3,0 3,1 81.43897 LENGTH-I 3.1 3,0 .2376603 LENGTH 5,2 5,1 .3319301 FACE 5,1 4,0 5,0 44.35585 AXIAL 0.0 3,1 3,2 81.38710 LENGTH-D 2,1 3,1 .2383753 LENGTH 5,3 5,2 ,3779644 FACE 5,1 6,1 5,0 40.69851 If ' AXIAL 0.0 4,0 4,1 82.07062 l.ENGTH-E 3.2 3,1 .2526047 LENGTH 6,1 6,0 .1970752 FACE 5,2 4,1 5,1 59.36916 we AXIAL 0.0 4,1 4,2 82.07062 AXIAL 0.0 1,0 0,0 84.26350 LENGTH 6,2 6,1 .2654663 FACE 5.2 6.2 5,1 52.97068 be FACE 1,1 0,0 1,0 71.20597 AXIAL 0.0 1,0 1,1 83.28597 LENGTH 6,3 6,2 .3203644 FACE 5,3 4,2 5,2 67.79066 thi FACE 1,1 2, 1 1,0 65.29889 AXIAl. 0.0 1,0 2,1 83.67240 AXIAL 0.0 1,0 1,1 82.02881 FACE 5,3 6,3 5,2 59.99999 FACE 2,1 1,0 2,0 6605975 AXIAL 0.0 2,0 1,0 83.58052 AXIAL 0.0 2,0 2, 1 80.7 8252 FACE 6,1 5,0 6,0 34.88121 FACE 2,1 3,1 2,0 61.43574 AXIAL 0.0 2,0 2,1 82.99787 AXI AL 0.0 3,0 3,1 80.44669 FACE 6,1 7,1 6,0 40.08534 A · FACE 3,1 2,0 3,0 60.31862 AXIAL 0.0 2,0 3,1 83. 36200 AXIAL 0.0 3,1 3,2 79.10660 FACE 6,2 5,1 6,1 45.48972 Se FACE 3,1 4,1 3,0 57.35055 AXIAL 0.0 3,0 2,0 83.46727 AXIAL 0.0 4,0 4,1 81.48806 FACE 6,2 7,2 6,1 55.29554 FACE 3,2 2,1 3,1 60.00000 AXIAL 0.0 3,0 3,1 83.17540 AXIAL 0.0 4,1 4,2 78.89616 FACE 6,3 5,2 6,2 56.10466 tri FACE 3,2 4,2 3,1 57.12850 AXIAL 0.0 3,1 2,1 83.15477 AXIAL 00 5.0 5,1 83.05105 FACE 6,3 7, 3 6,2 70.32347 II I FACE 4,1 3,0 4,0 54. 39701 AXIAL 0.0 3,1 3,2 82.74402 AXIAL 0,0 5.1 5,2 80.44669 FACE 7,1 6,0 7,0 31.34965 FACE 4,2 3,1 4,1 53.62160 FACE 1,1 0,0 1,0 71.58361 AXIAL 0,0 5,2 5,3 79.10660 FACE 7,1 8,1 7.0 40.67220 CU DIHEDRAL 1,1 1,0 172.5324 FACE 0,0 1,0 1,1 54.20819 AXIAL 0.0 6,0 6,1 84.34503 FACE 7,2 6,1 7,1 40.98321 Se DIHEDRAL 2,1 2,0 171.0111 FACE 1,1 1,0 2,1 57.96754 AXIAL 0.0 6,1 6,2 82.3 7243 FACE 7,2 8.2 7,1 58.11872 ler DIHEDRAL 3,1 3,0 169.5426 FACE 1,1 2, 1 1,0 64.06490 AXIA L 0.0 6,2 6,3 80.78252 FACE 7,3 6,2 7,2 53.68360 DIHEDRAL 3,2 3,1 169.3659 FACE 1,0 2,1 2,0 57.32457 FACE 1,1 0, 0 1,0 89.44366 FACE 7,3 8,3 7,2 78.98877 DIHEDRAL 4,1 4,0 168.5529 FACE 2.1 1.0 2,0 66.60298 FACE 1,1 2,1 1,0 69.93465 FACE 7,4 6,3 7,3 61.26167 CC DIHEDRAL 4,2 4,1 168.2212 FACE 1,0 2,0 2,1 56.07243 FACE 2,1 1,0 2,0 78.14680 FACE 7,4 8,4 7,3 88.52971 A A FACE 2,1 2,0 3,1 60.17371 FACE 2,1 3,1 2,0 61.03260 FACE 8,1 7,0 8,0 30.24383 St FACE 2,1 3,1 2,0 62.53881 FACE 3,1 2,0 3,0 65.02343 FACE 8,2 7,1 8,1 40.35929 A FACE 2.0 2.1 3,1 57.28745 FACE 3,1 4,1 3,0 55.29554 FACE 8.3 7,2 8,2 55.86596 FACE 2,0 3,1 3,0 58.04666 FACE 3,2 2, 1 3,1 69.40889 FACE 8,4 7,3 8,3 71.62383 Se FACE 3,1 2,0 3,0 62.40113 FACE 3,2 4,2 3,1 58.76529 DIHEDRAL 1,1 1,0 174.9689 wi FACE 2,0 3,0 3,1 59.55219 FACE 4,1 3,0 4,0 53.9439& DIHEDRAL 2,1 2,0 178.4242 FACE 3,2 2,1 3,1 63.99017 FACE 4,1 5,1 4,0 53.94395 DIHEDRAL 3,1 3,0 174.3684 CC FACE 2,1 3,1 3,2 58.00490 FACE 4,2 3,1 4,1 60.61734 DIHEDRAL 3,2 3. 1 177.4101 se DIHEDRAL 0,0 1,0 171.6929 FACE 4, 2 5,2 4,1 60.61 734 DIHEDRAL 4,1 4,0 166.3640 DIHEDRAL 1,1 1,0 174.4714 FACE 5.1 4,0 5,0 47.66244 DIHEDRAL 4.2 4,1 168.8965 DIHEDRAL 2,1 1,0 171 .'1439 FACE 5,1 6,1 5,0 55.03267 DIHEDRAL 5,1 5,0 164.2287 C · DIHEDRAL 1,0 2,0 171.7234 FACE 5,2 4,1 5,1 55.29554 DIHEDRAL 5,2 5,1 162.5781 SC DIHEDRAL 2,1 2,0 173.2939 FACE 5,2 6,2 5,1 65.02343 DIHEDRAL 5,3 5,2 162.9038 DIHEDRAL 3,1 2,0 171.7906 FACE 5,3 4,2 5,2 58.76529 DIHEDRAL 6,1 6,0 165.6335 C- DIHEDRAL 2,0 3,0 171.8081 FACE 5,3 6,3 5,2 69.40889 DIHEDRAL 6,2 6,1 163.7834 DIHI::DRAL 3,1 3,0 172.8261 FACE 6,1 5,0 6,0 45.27816 DIHEDRAL 6, 3 6,2 164.1306 Se' On this rage are DIH EDRAL 2,1 3,1 171.7525 FACE 6,2 5,1 6,1 54.19075 DIHEDRAL 7.1 7,0 167.0795 CU DIHEDRAL 3,2 3,1 172.8762 FACE 6,2 5,2 6,2 61.03260 DIHEDRAL 7,2 7,1 167,1492 Co Chord Factors for: DIHEDRAL 1,1 1,0 176.9902 DIHEDRAL 7,3 7.2 169.8046 various other DIHEDRAL 2,1 2,0 172.8942 DIHEDRAL 7,4 7,3 172.4355 Sli methods and DIHEDRAL 3,1 3,0 168.8103 DIHEDRAL 8,1 8,0 166.1374 112 breakdowns DIHEDRAL 3,2 3,1 169.1513 DIHEDRAL 8,2 8,1 166.4841 DIHEDRAL 4,1 4,0 167.4891 DIHEDRAL 8,3 8,2 169. 2092 CU DIHEDRAL 4,2 4,1 167.3106 DIHEDRAL 8,4 8,3 173.4652 Sel DIHEDRAL 5,1 5,0 168.7345 DIHEDRAL 5,2 5,1 168.8103 Cu DIHEDRAL 5,3 5,2 169.1513 CO DIHEDRAL 6.1 6,0 . 168.7447 Sta DIHEDRAL 6, 2 6,1 168.9111 DIHEDRAL 6,3 6.2 169.6921 r = radius b = base h = altitude Angle marker useful in measuring al l those wierd angles. Gel one at a drafting st are. Decimal Equivale nt, of Frac t ious 1/ 32 .03125 1/1 6 = .06250 3/ 32 = .09375 <0 1/ 8 = ,12 500 E 5/ 32 ,15625 u 3/ 16 " ,18750 (]) v 7/ 32 = .2 18 75 (]) '" .<: .... 1/4 = .25000 '" 0 E u 9/32 = .28 125 5/1 6 ,• .31250 - -0 11/ 32 = .34375 .<: .... u a 3/8 . .3 7500 c.<: ._ u c 13/ 32 = .40 625 .:::: E 7/1 6 = .43750 o ";::; 15/ 32 = .46875 '" '" c ::> 1/ 2 = ,50000 0 '- ,- "1J 17/ 32 .- .53125 '" co u 9/16 = . 56250 0'> - c 19/ 32 = .59375 ", ' - '" >- 5/8 = .62500 ",- _ 0- - '" 21/32 '" .65625 u - - ::l 11/ 16 = .687 50 E a >- 23/32 = .71 87 5 "'.D 3/4 = .75000 &-0 (]) (]) 25/ 32 = .78 125 s: c '" .- 13/1 6 = .8 1250 '" (]) '" "'.0 27/32 = .84375 ::J a 7/ 8 -- .87500 29/3 2 . - ,906 25 15/1 6 '" .93750 31/32 .96875 1 =1,00000 ·W 2nr TU 2 4 m 2 4/3Tl r 3 ,. , (j'h B G 0 • A 12 R " D ,. "ia 24 30 36 4S 54 2' 54 Conversion of feet to meters/meters to fee t : Area of t riangle 1/ 2 bh No te: by doubling t he d iame ter of a sphe re, the surface area is increased by a facto r of 4; the Volume is increased by a factor of 8. In any convex polyhedron, the number of laces (Fl , ver t ices (V) and edges (E} are related as fo llows: 1 ft = 0. 3048 meters 1 rneter v 3,28084 ft rr ='l.14 1:J926:> This form ula Can be used in che cki ng domes which are not complet e spheres: consider th e open bottom as a sing le face polygon; the number of sides eq ua ls the nu mber 01 members along th e per imeter of t he do me's f ramework, Circumference of circle Area of circ le Ar ea of sphere (skin) Volume of sp here I t _r.Ht IN rut 10 12 14 16 11/':1 '2 2lh 2VJ '2\h 3 Jlh 4 3 1 / 1 4 "" '!l 5 1h S 6 7 8 6 % 8 9 1/j 10'1:1 8113 10 11% 131h 10 ],2 14 HI 6% g 91; 1 10 12 14 16 13 1/ 3 16 IBlfJ 21 1/ 1 16'h 20 :l!31h 26'11 20 24 28 ]'2 25 J O 35 40 30 16 42 48 I ] I!, 16 21 113 30 3b 42 4K D. F DA Ec ..Q.. M BD e.E:.. DES M E;A SIlt I by.2 1 by 3 1 b, 4 ) by 6 J by I by 10 I b, 12 ? by 4 2 by ti zb, 8 2 br 10 2 by J2 3 by JO 3 bIll 4 b)" 4 6 by 6 e 1 Sl ANT RUlE WITH DESIRED NUMBER OF :2 REPEAT AT DIYISIONS ACROSS END BOARD ..----- ' , . ---------- " . ---; A :DLv:r:D..:I:N"U i:1r:J;o :t?rFQ?ie This holds true also in 3-0 space and may be used for calculat i ng cu t-orf dime nsions . Where a l i ne is passed parallel to t he base of a Iriangle at o distance Q Irom t he base then: Not e: See simple pocketbook Trig ts bles in biblio- graphy. I(-?C- ta, '10 0 f.. - D C • TWO 8R:ADS 4 TIE A L, Go r 3 DR'VE CE A·C APART EXACTLY HALF O" TAN 0' ---- ..... - lENGTH A·( PLUS ec -- ""--., ( -- _... r- d WC ' ... ....... ON IlAAOS. 5 WITH INStDE IT 5WING PE TO DRAW ELUf'SE " 2 "t,2 + c 2 .. 2bc cos A b 2 a 2 + c 2 - 2ac cos B c 2 a 2 + ,,2 - 2ab cos C TmG 1 ON A SOUARE. DR W GE OF SOUARE.CUT PAPEIf. MINOR OE'SlflED Elllfl S£ A·B cos " side 0; " hy potenuse c tan " = )- I" Side adiacen t b cot ..,( side - Jl-- I" sodeopposite a sec Y f/ side ad jacent b csc l y side opposi te a ECNT . .r: I' -'. - t--.;". _.__,( ... C/ i 2 MA.QX LeNGTH \ft MA JOR A)l IS B. TO C c 2 a 2 + b 2 180' 90" J ++ The t ri gonomet r ic f unctions may be lound graphi cally with the aid of t he f oll owi ng diagram : ,- COT ANGf Nl -'1 :> OBLIOUE TR IANGLE TRIGONOMETRIC FU NCTIONS B //j \ LL c / --- f>.. b RIGHT-ANGL ED TRIGONOMETRIC FUNCT IONS R ight-angled t r i gono metric [ unct io ns: The trig functions .ra the rati os of sides of r ight t r iangles. Func ti o n . p side ouoosite a., sin ....,. ....:.; ,.",.- hypotenuse c 5 Law of si nes: Leng lhs of t he sides of a triangle are proportional to the sines of th e anqles opposite t hem: " b c SiflA sin 8 sine Lawai cosines: The square of the l ength 01 a si de of a tr iangle equals the sum of the squares of the l engths of the ot her two sides minu s t wice the product of th ese two si des times tile cosi nes of the angle between them. s trut length Iebet s EX PANDED OCTAHEDRON FREQUENCY 4 EXPANSION · 1. 6 1800 LENGTH- t 0,0 1,0 .32694 6 AXIAL 1,0 76. 372 3 AXIAL 0,0 75.2881 LENGTH-2 1, 1 1,0 .4472 13 AXIAL 1,0 70.3873 AXIAL 1, 1 70.3873 LE NG TH-3 1,0 2, 1 .469976 AXIAL 2, 1 74.5005 AXIAL 1,0 72.0855 LE NG TH -4 1,0 2, 0 .552780 AXIAL 2,0 76. 7174 AXIAL 1,0 73.3400 LE NG TH-5 2, 1 2,0 . 536012 AXIAL 2,0 72.5012 AXIAL 2,1 70.9763 L ENGTH-6 2, 1 3, 1 .776522 AXIAL 3, 1 76 .90 18 AXIAL 2, 1 73.7454 LENGTH-7 3,2 3, 1 .577350 AXIAL 3,1 62.2539 AXIAL 3,2 72.2539 LENGTH-8 2,0 3, 1 .642227 AXIAL 3, 1 77.2756 AXIAL 2,0 75.4659 LENGTH-9 2,0 3, 0 .677009 AXIAL 3,0 80.7354 AXIAL 2,0 79. 1875 LENGTH-l0 3, 1 3,0 .454211 AXIAL 3,0 77.3557 AXIAL 3, 1 77. 0637 LENGTH- l 1 3, 1 4,2 .7332 14 AXIAL 4,2 79.4 716 AXIAL 3,1 78.8762 AXIAL LE NGTH- 12 4,2 4,1 .459505 AXIAL 4, 1 76. 7174 AXIAL 4,2 76. 7 174 LE NGTH- 13 3, 1 4,1 .679898 AXIAL 4,1 81 .8569 AXIAL 3, 1 81 .3987 LENGTH- 14 3,0 4,1 .60149 1 AXIAL 4, 1 80.4299 AXIAL 3, 0 80. 116 7 LENGTH- 15 4 ,1 4,0 .320364 AXI A L 4,0 80 .7825 AXlj\ L 4,1 80.7825 LE NGTH-16 3,0 4,0 514223 AXIAL 4,0 84.2726 AXIAL 3,0 840863 Hal f of t hi s str etched octahedron can be pu I toget her With the spherical 4·frequency oct ahedron on the opposite page. There are paper models of bo th 01 th ese oc t ahedrons on p. 124. Note: usc same vert ex lsb cts 'I S regu/itT al te r-nato OCTAHEDRON EL L! PTI CAL Using the octahedron means that there is on ly one pri nciple triangle involved, All the rest ilre the same, I t can be tr uncated perfect ly zentth-to-zeruth tu pr oduce " shape such as the paper model on p. 124 or to srt high l i ke t he agg. I n eit her case, t he octuhedr nnal fo rm has the dist ingui shmg abil i ty to f use to rect il in ear struc tures. All al te r nat e brua kdown is used, Nne af svrnmctr v 'e Ierne: ngle f rarnu 'o lted - vs to ! f it 21 tube st op block 2Y." mark Hif. 2// o Fig. 48 about %" support bloc k with V nat! guide angl e iron jaws Filf. J tube "lip s" ~ V groove ex ect depth 10 support flat tip v ise vise Fig. 4A lube F ' : ~ . 2A \ Cutting The tubes should be cut according to the chord factors plus 1 1/2". The chord factor gives the "center-of-hole to center-of-h ole" length, and there must be about 3/4" beyond the holes. Conduit comes in ten foot lengths. You get two struts from each length for making domes up to about 24 feet diameter . If you are making a smaller dome, try to size the dome to minimize wa ste. Perhaps you can get three st ruts from a length if you combine two A and a B or something like th at. Think before cutting. Make the cuts with a hacksaw or a tube cutter. We find that co ndu it cutting blades for a table saw are actually slower than a hand hacksaw. Use a 16 or 18 tooth hack saw blade. Hold the tube in a vise with a stop on the bench positioned so that a cut made right against the jaw of the vise will be the correct length. Keep the tubes in separate piles. Garbage can s make handy holders. Drilling To drill the first hole in each strut, cut a Vee groove in a 2 x 4 six inches longer than th e C strut, and clamp it to the drill press table with C-clamps. (Fig. 4A) Nail a stop block across the groove so that the drill hits dead center on the flattened tube tip and the edge of the hole nearest to the tip is about 3/4" from the tip. (Fig .4BI For 3/4" conduit a 3/&" bolt should be used. To facilitate assembly and absorb errors, the hole you drill should be 7/16 or 15/32". To drill th e second hole in each strut, measure exectiv the correct hole-to-hole distance (that the cho rd factor has shown to be correct} along the 2 x 4 and drill a 3/8" hole there. There will be a hole for A, Band C. In the A hole screw in a 7/16" bolt that has had its head cu t off with a hacksaw (Use a Vise-qrip or pipe wrench). Fi Ie the cut-off stump to a sort of point so that the first hole already drilled in the tips of the struts can easily fit over it . Again clamp the 2 x 4 to the table so that when the tube has its first hole impaled over the cut-off bolt pin at the Fig. 1 Squashillg Flatten the tube tips 2 1/4" from the ends, by squeezing them in a vise . A big vise . Small "home workshop" vise s will break. (BASCO makes big ch eap vise s]. Squeeze the tubes horizontally in the vise . If the jaws are not big enough, sti ck on small pieces of angle iron (t 1/2" wide) with pu tty to enlarge the squeezing are? of the jaws. (Fig .2A) Make a mark on the jaws 2 1/4" in from the edge as a quid e for depth o f squeeze. (Fig .2B) Insert the tube; holding it perfectly horizontal and centered in the jaws vertically so that all of the tube will get squashed. Oil the vise screw threads to make the turning eas ier. It will probably be necessary to use a "persuader" pipe on the vise handle about 2 feet long, but be careful not to over- strain the vise . EMT tube has a weld running the entire length of it . If th e tube is positioned in the vise so that this weld co mes at the very edge of the squashed tube, it will split. To prevent this, the tube should be positioned in the vise with the weld at 2 o'clock . (Fig 2A ) Splits along the edge of the flattened tube are rejects. Splits in the middle of the flat are undesirable but usable. To squash the opposit e end, eyeball the flat you have just made and insert the other end in the vise as nearly in the same plane as you can. This end will automatically have the weld positioned correctly, so you don't have t o worry about il. The tubes should be squashed as flat as you can manage with your vise, but a smal l amount of "lips" is acceptable. (Fig.3} You can flatten these with a hammer later. If the dome is to be skinned with plastic film or fabric that touches the frame, file or grind off the sharp corners of the tips at this time. These tips will rust, so they should be painted silver with rust -prevent inq paint. Th e vise-squeezinq bit is tiresome. Try and wangle someone into doing them ana press in a machine shop. Hammering tips flat without a vise results in poor fit and a generally crappy appearance. lUi Frame Dome Tube framed domes have many advantages over wood for certa in uses. They use mineral mater ials instead of killing trees, and when obsoleted their scrap can be largely recycled. Tube frames are well suited to flexible non-Ioad ·bearing skins and are the simplest way to make a "sky break" with no sk in at all. They can be skinned in several ways: a fabric tent suspended ins ide the frame by rubber bands; a fabric tent applied to the outside of the frame and resined ; fiberglass or metal sheets; Plexiglas; or even plywood . Transparent or translucent panel s can be made from vinyl and inflated . The frame can be covered with a net or mesh and foamed or ferrocemented. Many other skins co uld be tried using the basic principles outlined below. VITAL STATISTICS (for Bubble Dome) Geometry: 3-frequency geodesic, 5/8 sphere, ico sa-alternate breakdown, vertex zenith Diameter: 20' Weight {not including floor) : 600 Ibs Volume: about 2600 cu It Floor area: 314 sq It THE FRAME Any suitably strong tubing can be used, but the cheapest and eas iest to get is " EMT" electrical conduit. It is easy to work with and is plated, so painting isn't necessary. 1/2" is not suitable for any domes that will be subjec ted to heavy weather co nd it io ns, but it is useful for indoor structures and small (up to 14 feet diameter ) domes. 1/2" conduit will bend if climbed on . 3/4" is best for most uses. It wholesales for about 9li/ft . Using the chord factors, you can use 3/4" conduit in triangles whose sides are up to 4 1/2 feet long . 4 feet is maximum where there will be snow loads. This will result in about a 24 foot maximum diameter in 3-frequency. For larger domes you will need bigger tubes or a higher frequency. Bigger tubes are hard to squash! Think first. (See "What Size?" p. 49) I.) use a lis to . One • staples .e, ) nail e sure lone to un ison. are arks 21 tube stop block 2V." mark Fig. 2fl o Fig. 48 about %" sup p or t block with V nail g u i d ~ angle iron jaws Fig. 3 lube " lips" ~ V groove exact depth to support flat tip vi se Fig. 4A tube Fig. 2A CUlling The tubes should be cut according to the chord factors plus 1 1/2". Th e chord factor gives the "center-of-hole to center-of-hole" length, and there must be about 3/4" beyond the holes . C o ~ d u i t comes in ten foot lengths. You get two struts from each length for making domes up to about 24 feet diameter. If you are making a smaller dome, try to size the dome to minimize waste. Perhaps you can get three struts from a length if you combine two A and a B or something like that. Think before cutt ing. Make the cuts w ith a hacksaw or a tube cutter. We find that condu it cutting blades for a table saw are actually slower than a hand hacksaw. Use a 16 or 18 tooth hacksaw blad e. Hold the tube in a vise with a stop on the bench positioned so that a cut made right against the jaw of th e vi se will be the correct length. Keep the tubes in separate piles. Garbaqe cans make handy holders. Squashing Flatten the tube tips 2 1/4" from the ends, by squeezing them in a vise. A big vise. Small "home workshop" vises will break. (BABCO makes big cheap vises). Squeeze . the tubes horizontally in the vise. 1f the jaws are not big enough, stick on small . pieces of angle iron (1 1/2" wide) with pu tty to enlarge the squeezing area of the jaws. (Fig.2A) Make a mark on the jaws 2 1/4" in from the edge as a guide for depth of squeeze. (Fig .2B) Insert the tube; holding it perfectly horizontal and centered in the jaws vertically so that all of the tube will get squashed. Oil the vise screw threads to make the turning easier . It will probably be necessary to use a "persuader" pipe on the vise handle about 2 feet long, but be careful not to over - st rain the vise. EMT tube has a weld running the entire length of it . I f the tube is positioned in the v ise so that til is weld com es at the very edge of the squashed tube, it will split. To prevent th is, the tube should be positioned in the vise with the weld at 2 o'clock. (Fig 2AI Splits alonq the edge of the flattened tube are rejects. Splits in the middle of the flat are undesirable but usable. To squash the opposite end, eyeball the flat you have just made and insert the other end in the vise as nearly in the same plane as you can. This end will automatically have the weld positioned correctly, so you don't have to worry about it. The tubes should be squashed as flat as you can manage with your vise, but a small amount of "tips" is acceptable. (F ig. 31 You can flatten the se with a hammer later. I f the dome is to be skinned with plastic film or fabric that touches the frame, file or grind off the sharp corners of the tips at this time. These tips will rust, so they should be painted silver with rust -preventing paint. The vise-squeezing bit is tiresome. Try and wangle someone into doing them on a press in a machine shop . Hammering tips flat without a vise results in poor f it and a generally crappy appearance. F(".l Drilling To drill the first hole in each strut, cut a Vee groove in a 2 x 4 six inches longer than the C strut. and clamp i t to the drill press table with C-clamps. (Fig.4A) Nail a stop block across the groove so that the drill hits dead center on the flattened tube tip and the edge of the hole nearest to the tip is about 3/4" from the tip. (Fig.4B) For 3/4" conduit a 3/8" bolt should be used. To facilitate assembly and absorb errors, the hole you drill should be 7/16 or 15/32". To drill the second hole in each strut, measure exactly the correct hole-to -hole di stance (that the chord factor has shown to be correct) along the 2 x 4 and drill a 3/8" hole there. There will be a hole for A, Band C. In the A hole scr ew in a 7/16" bolt that has had its head cut off with a hacksaw (Use a Vise-grip or pipe wrenchl. File the cut -off stump to a sort of point so that the first hole already drilled in the tips of the struts can easily fit over it. Again clamp the 2 x 4 to the table so that when the tube has its first hole impaled over the cut-off bolt pin at the De Frame Dome Tube framed domes have many advantages over wood for certain uses. They use mineral materials instead of killing trees, and when obsoleted their scrap can be largely recycled. Tube frames are well suited to flexible non-load-bearing skins and are the simplest way to make a "sky break" with no skin at all. They can be skinned in several ways: a fabric tent suspended inside the frame by rubber bands; a fabric tent applied to the outside of the frame and resined; fiberglass or metal sheets; Plexiglas; or even plywood. Transparent or translucent panels can be made from vinyl and inflated. The frame can be covered with a net or mesh and foamed or ferrocemented. Many other skins could be tried using the basic principles outlined below. VITAL STATISTICS (for Bubble Dome) Geometry: 3·frequency geodesic, 5/8 sphere, icosa-alternate breakdown, vertex zenith Diameter : 20' Weight (not including floor) : 600lbs Volume: about 2600 cu ft Floor area: 314 sq ft THE FRAME Any suitably strong tubing can be used, but the cheapest and easiest to get is "EMT" electrical conduit. It is easy to work with and is plated, so painting isn't necessary. 1/2" is not suitable for any domes that will be subjected to heavy weather conditions, but it is useful for indoor structures and small (up to 14 feet diameter) domes. 1/2" conduit will bend if climbed on. 3/4" is best for most uses. It wholesales for about 9d/ft. Using the chord factors, you can use 3/4" conduit in triangles whose sides are up to 4 1/2 feet long. 4 feet is maximum where there will be snow loads. This will result in about a 24 foot maximum diameter in 3-frequency. For larger domes you will need bigger tubes or a higher frequency. Bigger tubes are hard to squash! Think first . (See "What Size?" p. 49) ;to sea One taples nail sure one 1 on. re ks / TUBE' External An ex ter tube [rar: The skin resined. a weight E will resul that rniqt Thin She The fram will also i th e tube edge of tt position, the edges water. Pr start at t ~ t riangles [ is over w ~ edge will (Fig . 9) I sively, a c that is fi e the panel : supply stc to the tut be roll ed , cheap.) [ stv ro toarr translucer fiberglass way to in: interior, a Plexi"tlas s Domes car and still bt metal to ri The strips and bent i' at th e hub the Plexigl expansion A second l Dum on b, Plywood s Plywood ~ woods wit th e plywo dirt. Ther longest), s have to ca as it can tl fr om pane all seams are act ually straight bu t curve in use seam ope ned Inner tu ll e band kno tred u noern eath through 4" plywood di sc th ,s end hook ed to vertex F ( ~ . 8B welt scam liD seam #4 grommet in canvas sew 2 rows ---1:+ A tbree- frequency dome doesn't sit flat on the gro und, but rather sit s on five points. You can either block up the ot her ten points 0 1' lower the dome on the floor so that the five low po int s are below platform level. In an y ca se, it will be necessary to support ALL the poi nts if the dome is to be reall y st ro ng. Remember that wind lift is the largest load your dome will take, other than heavy snow. Assembly Assembl e the frame start ing at the bottom or tor (and lift it as you go). as you prefer. For larger domes start at the bottom. Bo lt t he botto m co urse tightl y , but only st ick a bolt through the ne xt layer so that it will be easy to add to. If you ha ve drill ed accurately the do me will go together very quick ly using two stepladders or a sca ffo ld . Un to about 30 teet . no supports will be needed . Remember that the removal of any strut ser iou sly weakens the dome! Removed struts, as for doors, etc ., MUST be compensat ed for with extra bracing that maintains the angles! Before adding any skin of any description, it is AHSOLI ITELY NEf:ESSARY to fasten the dome secu rely to the ground, or it will take off and fly remarkably well. A ligh tning ground rod is also ABSOLUTELY "I'/o:CESSAIt Y! Connect to any hub bolt. SKINS Suspended Skin The tube dome can be skinned with fabric by sewing togeth er triangul ar panel s that are somewhat smaller than the chords of the tubes but ar e in the same ratios. If the clot h is wide enough, several panel s can be made together rather than making all the triangles. The pieces should be sewn together using a double needle industrial machine and Dacron thread made specifi cally for tent making. Materials such as mu slin and canvas can be used . For permanence, Hypalon or Acrilan ar e best but expensive. Th e seams can be "fin seamed" if they're hemmed first. This is a drag, and the be st seam is the "welt seam" unless a special tentmaking machine is available. (Fig.8A) A home sewing machine will not work well unless the dome is sma ll o r the fabric very light. A para - chu te will fit over a qeodesi c frame, but will be diffi eu It to waterproof and wi II not fit perfectly. OK for summer shade. These skins can be attached to the ringbolts with innertube bands cut with tin snips, from tru ck tu bes. Car tubes are too weak . Attach with #3 grommets at vert ices and short knotted ro pes through the rubber ba nd s. (Fig. BBI 3 2 4 I-i'g. ,r, TUBE FRAME DOME continued apply pressu re t owards p in wh ile drill ing 22 (fr om ou tslde dome look ing in) numbers are stac kiog order Fi g. 6 other end of the 2 x 4, the seco nd tip will be under the drill at pr ecis el y the co rrec t place. ( Fig. 5) Start th e drill and lightl y touch it to t he t ube. Stan the drill and measure f rom the center o f the first hole over the bolt to t he dr ill scrat ch to check if it is th e co rrect hole-to-hole distance. I f it' s o.k ., you can quickly drill oil th e seco nd holes accura te ly in the A st ru t s. Repeat for the Band C st ruts (moving the pin), checking each time for accuracy before drilling th e wh ole bat ch . Mark the table and th e 2 x 4 so you can check if the jig is moving as yo u work (which is disa ster) . Hold the st rut with pr essure against the pin to t ak e up any slack. Thi s eliminates most error . Careless drill ing will res ult in maddening misaligned holes during assembly! Bt>lItling Tip , The t ips can now be be nt to approximatel y the angle that they will hav e in the fin ished dome frame. Do thi s in the vise by inserting the flattened tip in the vise and bending th e tube t o a stop block nailed to the table. All ow for " sprinqba ck " by bending a bit furth er th an seems right. A few tries will sho w th e proper place . For 0 3·frequency dome, bend the A st rut s to 10 112 0 and th e B ane! C strut s t o 12° . Accuracy in thi s bend is not important. (Fig. 6) Ac curacy in drilling t he holes is very important unless you like lumpy domes assembled by beating them with a sledgeha mmer. I f flattened t ips have split across the hol e, they shou ld be bent towards the split side so spl it will be inside th e dome. Bolts The bolt s should be long enough to go throu gh six flattened tips (remember that th ey probabl y vary in flatness. Assume they are all as thi ck as the worst one you can find) and two washer s (one on each side of the st ack). plu s enou gh to eas ily get th e nut unto. I t is wel l to have them an inch longer than that , or even longer -which is handy for attaching things to the dome later . The bolt head s will be inside th e dome and th e nut s outsid e. We use ringbolts (with the ring insid e) so th at we ca n easi l y hong things fro m the inside o f the dom e. Be sure to get th em with enough thread near the neck to tight en th e stack of tips properly . Tip s sho uld be stacke d in an o rde r t hat makes an y given triangle as level as pos sibl e. (Fig. 7) Thi s will vary with the tvpe of skin that will be used, but in any ca se, make each st ack of a given type (hex, pent, irreqular hex) the same way. This will make re-sta ck inq ea sier if that sho uld prove nece ssary, and will make the dome symmetric. Different tvpcs of skin will require different stacking. Think it out, and make a test section before charging. 23 In any of these domes, there are many small details that you must work out for yourself as you go, as there is not a great deal of collected experience to guide you. Th is infor mat ion has been pr oven to work bas ically. Try new ideas with test sect ions f irst. . InOated panellikin A way to get a transparent dome that is insulated to an extent is to make the panels from inflated vinyl " pil lows". Use 20 mil transparent or translucent sun resistant vinyl, and have the pillows seal ed by a pro in a big city. They'll be about $3.00 a panel which co mpares well with F ilon and even plywood aft er you pay for the pai nt . These panels will last about three year s, but might last more or less depending on sun condit ions where you live. They are cut to have the seam we ld along the inside of each tube edge, and must have a 2" border outboard of the weld for clamping. These pillows are fastened to the tube frame by impaling them (shingled) over the vertex bolt tips and th en clamping them to the frame with half-round strips made from 3/4" PVC irrigat ion pipe split lengthways in half . Have a sheet metal screw every 6" . (A #8 screw is qood] This will mean a lot of screwdriver work and drilling, but th e result is nice. Blow them up with dry air or nitrogen to prevent condensation. (Fig. 12) See Bubble Dome chapter. Fig. 10 poi nt d own panel ove rlaps po int up panels. f' (/? 9 Fi,l/. I;: look inq from outdoors External skin An external sk in can be made from triangles the same si ze as the outside of the tube frame as measured in several sample places. Be su re and allow for shrinkage! The skin can be sewn to the frame here and there. These sk ins will leak unless resined. Apply a heavy coat of flexible resin or foam . Resin can be applied with a weighted paint roller with a hinge in th e stick to get you over the hump. Thi s will result in a translucent watert ight dome. It will "boom" if thumped in a way that might drive you crazy if wind co ndit ions were wrong. Thin Sheet Stock Skin The fr ame can be skinned with fiberglass sheets such as "Filon" . This method will also apply to sheet metal. Cut the sheet s into triangles about 3/B" larger than t he tube frames. (With a 3/4" tube frame, cu t the sheets as large as the outside edge of the tubes in all cases.l Drill through th e sheet tri angles while they are in position, and rivet them on with POP rivets, overlapping the triangular sheets at the edges and particul ar ly at the hub bolts in a shingled manner so as t o shed water. Putty around bolts with DUM-DUM (see below). This means that you st art at the bottom and apply the "point up" triangles first , then the "point down" tri angles of each course. Where the struts are vertical it doesn't matter which edge is over which. I f the panels have been cut the right size, the overlap along each edge will be about 3/4" and in most ca ses this will be enough waterproofing. (Fig. 9) If a te st panel shows that there will be a gap th at wi ll admit wind exces- sively, a caulk could be applied between the panels before riveting. Use a caulk that is fle xible and sun resistant and that will not attack the panel material. If the panels buzz again st the tube frame, get a bo x of "st r ip-caulk" at an auto supply st or e (also referred to as Dum-Dum} and roll it into little ball s and sti ck it to the tube frame between each rivet . Thi s will dampen vibrati ons. It could also be rolled out into long thin snakes and used as caulk between the panels. lit's cheap.) Domes skinned thi s way can be insulated by applying one inch or th icker styrofoam panel s with mastic t o the inside surface, for metal domes, or clips for translucent Filon. Use fire retardant foam. Th is will stil l let a bit of light through fiberglass panels, but it won't be really very bright . It's about the only practical way to insulate suc h domes, because you won't have to make another skin fo r the interior, and this ends up costing less in the long run. TUBE FRAME DOME continued before assembly foi' g. II skin Domes can be skinned with transparent Plexiglas. This cannot be insul ated, remember, and still be transparent. It is very expensive, and it requires an ex t ra st r ip of sheet metal to rivet it to, as you can't overlap Plexiglas like you can fiberglass or metal. The strips ar e about 1 1/2" wide 20 gauge and are POP·r iveted to the tubes first and bent into a shallow V about 14 0 lengthwise. These strips must be "shingled" at the hubs, and puttied. Then rivet the Plexiglas to the str ips. Make the hol es in the Plexiglas oversized le.q. , T/4 " for a l /B" rivet). This will allow for the huge expansion and contraction of Plex iglas wh ich would otherwise break the panels. A second strip, also shingled at the hubs, goes over each joint with caulk or Dum- Dum on both strip surfaces. (Fig. TO) Plywood skins Plywood panels can be attached with "one hole clips" and 1/4" bolts. Cut the ply- woods with their edges exactly according to the chord factors of the tubes. Paint the plywood panels and let them dry thoroughly. Paint must be free of rough dirt. Then tape the joints with 2" wide weatherproof electri cal tape (black lasts lonqest], shingling it at the hubs and dum-dumming around th e bolts . You will have to carefully think out how you stack the tubes if you are using plywood, as i t can take just so mu ch bending. (F ig. 11) Try to avoid large "stair-steps" from panel to panel that would make taping difficu lt. . rI are t h icron e le ts. at pport largest bolt . rsten ifer. a curat elv about iously .r with Next we plan to try an all -openinq one, and it may well be a complet e sphere wh ile we are at it. We are also wor king on a production version with many improvements including pop together assembly without rivets or screws. And it's just the beginning . .. AnvwlIV, we decided to use a tube frame. We made a radial fl oor, as it was more economical for this size dome. We chose the 20 fo ot size because it was the largest dome we could make in three frequency, due to the maximum si ze of vinyl available. We decided to limit our design to three frequency, because a four frequency would use much more tube than was needed for strength . In the photographs you will see that the bottom course is skinned with Filon fiberglas riveted on. This was done because we intended to have a second floor at the level of the top edge of the Filon, and store all our stuff underneath the floor, accessible through a variety of trap doors. The translucent Filon would hide the stuff from people outside the dome, but would let in light. The idea was that when yo u came into the dome the entire thing would appear t o be empty and you coul d cal l into being any trip desired, such as a bed, th at was stored underneath. This way, the character of the pl ace could be instan tly changeabl e rath er than the usual home thing of having your house a sort of museum of mementos from your Mexican trip or funky old immobi le and ulti - mately bor ing relics . Sadly, the dome proved too small for this play, but we store every thing under the bed and can st ill have a relatively usef ul changeability in the space. Certainly you notice people more in our dome, at least onc e the unfamiliar- ity of the dome itself has worn off, which it does right away because we made th e shell purposely bare of all special charact er. The dome itself is only a with in-ness inside which we do our thing of that hour. We would never use the Filon again in th is way. I t booms in wind, and condensati on f orms on the inside in certai n weath er conditi ons. We also don't like the way it looks. Assembly was easy up to a point. As a test, we did the entire thing ourselves to make sure two persons coul d do it al one. The floor took one day . The frame went up in three hour s. HOWever, the bubbles, installed flat, took some 1300 sheet metal screws and three days. Doing it wrecked our arms and we had no feeling in our fingers for weeks afterwards. Next time POP rivets. We had some l eaks, part i cularly in the top pent opening. Five t ries and two weeks later we finally achi eved compl ete water- proofness even in violent storms. The waterproofness is not dependent on caulk, and the flexibility of the bubbles allows for expansion and contraction. We pumped them up from a tank of nitrogen in about an hour of deli cate hissing and one near explosion. We finished it off with an insulated rug and a water bed on high enough legs so that we can store all the stuf f we aren't using und er it. There is a fenced off area near the " door" f or taking off shoes so the dome won't be indistinguishable from the muddy fields around it. It is r eally nifty to li e there bobbing in our body-temperature water bed and be able to seestars, trees, moonl ight, birds, frost forming, snow, rain pelting, and, occasionally, spectators. The many vents keep it reasonable in hot weather. As with all domes it's easy to heat. The main thing is the super feeling of being almost outdoors. No roof , no womb, no hiding pl ace. Just weather out there and you inside. For country living, this is IT. Th e thing just feels super good to be in. It's hard to imagine getting mad in one. flat BV JIIV & Kathleen How would you make a dome that was completely tr ansparent and st i l l insul ated? Glassand other transparent sheet mater ial would have to be used in a double layer to get an insulation effect . That would be expensive, hard to accompli sh, and trapped moi sture would condense between the layers. There would also be severe seal ing probl ems and possibly even darnaqe to the skin arising from expansion and cont ract ion. About the same time we were thinking about these problems, we vi si ted Philo Farnsworth and saw models of his proposed spherical dwell ing on a pedestal, wh ich featured circul ar inflated plastic windows. We looked up Vinyl Fabricators in the Yell ow Pages. Pacific Vinyl Products in San Francisco made us a t est triangul ar inflated panel from 20 mil sun resistant vinyl electronically sealed at the edges and equipped with an air valve, We tortured this panel with thrown bricks and sharp heavy sti cks, but it didn't pop . We exposed i t to heat and cold with no serious eff ects. Fire, however, did damaqe the vi ny l even tho ugh it will not suppo rt burninq. For this reason we decided to inf lat e the panels with inert nitrogen; i t would t end to extingui sh any flame puncturing the ski n. Dry nitrogen also wi ll not· . , permit condensati on inside th e bubble. Fli<. J As we work ed with the test panel we discover ed that though it was a perf ect t r iangl e when fl at, it became invo luted into the shape seen in Fig. 1 when inflated. Th is "tendency" turned out t o be somewhat more than 200 lbs. at a mere 3 lbs. pressure! This ruled out any possibility of stapling or battening them to wooden frame members. We tho ught that wood was not appropriate anyway . As we war ked on the design it became apparent that the bubble panels trying to deform th e fram e could be a great advantage. Each bubbl e would be balancing the other bubbles al ongside so there would be no distortion of the st rut s. But there wo uld be a huge force trying to squeeze the dome in; that is, the struts would be uncler a great compress ion load. We thought that this would nicely bal ance the lift ~ l e n e r a t e d by win d, as this load is about 2 1/2 times the wind force push load. This turned out to be the case. Our dome weighs 600 lbs, but is absolutely solid in high buffeting winds. We worked out a clamping strip to hold the bubbles to the tubular struts (seephoto, page 22) , and also a method of opening the entire top pent by means of springs. We plan to make a dome with all the panels opening in thi s manner . I magine being able to open the whole thing by releasing a few ropes! A dome made this way would sacri- fice the pre-compressing feature that we just discussed, but this cou ld be compensated for by using heavier tubes. $ 160. 00 65. 00 225.00 5. 00 20. 00 96.00 30 .00 26.00 5.00 20.00 _12.00 $664.00 48 .00 35 .00 5.25 84.00 75:00 14.00 18.00 9.00 42 .00 say. . -, QQ 350 .25 about Total cost When you l ean back, the k not holds. To move, you take wei ght off t he k not. When you get t o desired position, l ean back , and you can work with bot h hands f ree. It' s a' stra nqe sensat ion-you'll gradually learn to trust th e r ig. It works best t o start at t op, and work your way down: you get so you learn the amount of reli ef needed t o descend , and soon you're wal king up and down on th e domes kin. Have care not t o slip feet up-head down; you'll fall out of the harness. <0 90 (- -- .;> Pod Dome 15' 5 1/ 2" di ameter, bent -over plywood dome Floor urea: 185 sq ft Volume: 7 Bubble Dome 20' diameter 5/ 8 sphere vinyl pill ow dome Floor area: 314 sq ft Vol ume : about 2600 cu ft Flo or con dui t vi nyl pillows infla ti on (ni t rogen) spri nqs, bolt s, nu ts, etc . Filon panels alum angle PVC pipe for cl amps lightning rod mise. Fig Newtons 1 1/8" tongue and groove plywood for fl oor, 6 sheets @ $8 mi sc. 2 x 6 and 4 x 6 joists and girders, 4 x 4 redwood posts, approx seven concrete pi ers @ .75 12 sheets 1/4" x 4' x 12' ext . grade pl ywood @ $7 one bubble skyl ight and one bubble window Pl ex iglas for windows rubber moul ding for di tto 3 rolls bu il di ng paper 3 squares #3 red cedar shi ngles @ $ 14 nail s, staples, bol t s, mendi ng str aps and misc. hardware, L oop 1 It' s best to take loop one around again and come through loop two again-for double protecti on. Thi s kno t wi ll sl ide when no pressure (weight) is placed on it , but will t ighten, and hold y ou as soon as you put your weight on it. Carabinier with safety l ock : Harness: You can buy a ready -made harness or make one out of about two yards of nylon st rap from a mou ntain climbing shop. The ends are tied t o make a continuous loop. To get into it, hold the ent ire loop horizontally behind your assand bring the loop ends together in front of your crotch. Hold them with one hand and reach back between your legs with your other hand to grab the lower li ne. Pull thi s l ine between your legs to meet with the ot her loops in front. Hook th e carabin ier through all three loops and wiggle the whole business up to waist level. The 6" or more par t is what you throw t he rope over . Lo ok in mountain cl imbing catal ogs for the equipment : a good 1/2 " nylon rope-Goldline is o.k. ; Edelreid Perlon i s better . About one yard of a slightly small er diameter rope (or ny l on strap). Ti e ends together t o f orm loop li ke thi s: I I often th ink of a dome as a boat haul ed as hore and t urned upsid e down. like a shi p, a dome has a ma st . The mast goes at top center, proj ect s 6" or more abov e th e do me, and (J ives you some- thing to throw a rop e over when climbing on t he ex t er ior (wh ich will be often) . Bill Woods t aught LIS t hi s met hod. He has $ 100 invest ed in cl imbing equi pment - i t ' s a 100 9 W"V down. t:J»c : -- - We made our masts l ike th is: : , 6'· or rnore 5 hol es : fo r st raps 00 _v __ --"" 2 5/ 8" '* . -. _. . . - If-11''-'''II $110.00 124.00 $234. 00 160.00 240.00 25.00 15.00 35.00 20. 00 35. 00 25.00 20.00 30.00 $9 29.00 -- I \ I> ..... I \ I PIlJv.. .......... , I CQM.ll11ll!1tI .. 01 I € .4 SMltJfd F<>R / -"\ --- \ '" 30SHEm \ \ -"1 r 1&3ie ee1.(.8/T \ L fldt fp:1 llOtA'S , -:» ;'" 0 l fi --;, "" 'JI. • -- - - - - - - • I -. / r '1"- - I PLYOoME I t l .It: . , / " 0 1 Vi Pr. .." Z. PT. .. I *' ,t . I I I .............. . ! , .. .' 1 ' 4' . I 1/ Total cost Paci ti « Dome 24' diameter 5/8 sphere plywood dome Floor area: 452 sq ft Vol ume: abou t 4500 cub ic f t Floor fr ame pl ywood Floor cost struts: total about 800 l in It @ 5380/ 1000 Bd. It pl ywood: total about 30 sheets @ $8 nails hubs str aps, buck Ies vinyl caul k pain t wood for window batts misc. insulation ydomes We've never built one, but i t looks as i f leakage would be a maj or pr oblem. You should pr ime and paint plywood carefully bef ore assembl y, incl uding edges and bolt holes, and either cauIk or use a neo - prene washer where bolting t ogether. Window details should be carefully consid- ered. Simpl e, cheap, qu ick t o build, plydomes are made of bolted-t ogeth er sheets 011 /4 " or 3/8" plywood . As with the Pod Dome, there is no separate internal framework. Ply · wood is shingled so water will run off . Try it in model form first . Seeal so Fuller's patent on plydomes. People from Canyon, Cal if. sent us these plan s: 5/8 Here are cost s for 1969· 70 of t hree di f f erent types of domes we have buil t . Mnterials ontv-sdoes not include labor. Cost s per square fo ot are decept ive, especially since in domes you usuall y build lo fts , and ar e not con f ined to the fl oor. Costs per cubic fo ot are a far better measure of value. Tot al cub i c leet in a dome depends upon what por t i on 01 a sphere you build: I n y ,e Id· iere 19- .vas here ne 'an- co :ol m . It .y 1 Iso ost I one les f my of was , of I to ,of ing . $14 s . • • vJ tV! lnt:o (use, tn, n "Stuff) or score. the. ?tUff) fr-orn ICOS1L -top vertex By uacinq these pee ls you can make t he models pictured. Trace t he peel five times for the icosa hedron based domes, fo ur t imes for th e oc tahe d ron bused domes. Cut out, incl ud ing t he gathering angles, t hen t ap e the gmher ing an gles toget her to for m curved sections. Then tape al l th e sect ions toge t he r t o make t he mod el. To dr aw t um pl at e of your own, f igure t he lengths fr om chord fac to rs, t he n d raw t he various lengt hs wit h a com pass, leaving o ne g(lth er ing angle around each vertex , e pe€'\ this IS 13h iccsa., ? " " .. '- .. " d in cut the ngles { e iving ... ?pcn d I I , TI lI1 um iet her .. ft 11) ' v ~ x '12 sphere \ .. " ,/ \ ". ! ' / , i ; I / , ; , .' r . '. / / \ I ,I c : . ~ t \ ,. \; ~ .1\ ,/ \ I ' , \ v • t • u ' sau 5 ~.~\.0. i> .l 'lpS and eornolete the :3 v/lIy gr id.~ I2 ~ A ~ -n connecte d with qrc at rnplete th e 3 ''1'1 <1. It is deve lop ed wit h lesser c ircl e as we ll ClS ' so tha t truncation m ay be don e w ithou t req u ir in g specia l cle men ts . "I relat ive di f f.with th e parts ch os en as equ al ar c di visions o f the Each point o f subdivi sions is then co n n ec t e d w it h line segments perpendi cular t o their res pe ctive prin ciple side th u s givi ng a 3 ·wa y grid co m p rised o f equil ateral and right trian gles.. The e leme nts jo in ing t h e t rans lated vert ices fo rm the chords o f iJ 3 'waY great c irc le gr id. Me t ho d 3: pherica] polvhedral t rian gle is subd ivide d into a low freq ue ncy subdi visio n. - A /" I . Du e t o lhi' me thod of subd ivis ion. Due to t he le ss sy m me t r ica l of thrs meth o d n is use d prim a rily fo r sma ll f requ e ncy struc t u res ._ - I I )-. The e le men ts con nec t in q th e t ranslate d vert ex lorm th e chor ds o f a 3 'WilY grea \ circular grid ... B G 8v (Vote.. In qe no ret [h e triaco n breakdo w n (Class 1/ ) is b ette r for large domes b eceuse t h e n u mher of ditier ent stru t leng rhs in c r eases arithmetically w it h the tr isco n (i. it is ting with a \n1<11I freq ue ncy and t he n su bdividi ng furth er to th o mcv by foll ow iuq u geo me trica l p rog res si on as pe r exa m p le: * (O. .. O < J.I me t hod s 1·3 allow inq for t ru ncat ion wi thi n th e I of the soherlcs l fo rm. " .> " 1 "~' ~" t' . No te: Al ' 1 2 Method 4 : { Th is m et h o d IS oesics tt» the seme as m ethod 3 excep t that instea d o f di vidino sid e AS of the right t ria n yl e w it h the eoue! arc: d i visions . (!q ultl arc di vis io ns o f c fI .1'1:1 chlMn as eq ual tir". Upon co m p letion o f t he co nnections cl grid is cre a te d. . ~ .' .m.~ r " r.. f circle. D ~ .'1 1 line se gme nt inte rsec ts at a n umbe r of p oi nt s wh ich division.o) I ivision on ('~ t'I pl'inc iple side of t he PPT ar e co n nec te d w it h lin c seqme n ts d ra. ~es in ed ge lengt hs a re greate r t han anv of t he ot he r To co m p lete t he 3 . ' . c. si de AC is dt vtd cd. J.d/ e d o /lof he on the same-g t13l1.\> // diV ided int o n ' reQlJllncy . \3) Not e: 'A B . 0 ..lple sides.---- \f} \ Not e : AT f IT T he po in ts of su bdi v ision on cac n p rinci ple side o f the PPT a re con necte d w ith line se gme n ts similar t o met ho d 1.pective side>.. ' I (0/0.. 0) o f the poly he d ro n . gI ven the 1l0W s /d r t ln g po. S Ea c h ve rte x on th e PPT is th e n t run slated o n to t he su rface o f th e c irc umscri be d s p here alo nq a lin e p assinq th rou gh the res pect ive ve rt e x an d th e o r igin (0 .. rhc line se gm ents a re not perpendi c ul ar to th cir res pect ive si des ./ I """"l /j5 I \:.. " 1 L~ ~: \. 0 ) o f t h e po tv hedro. ' '0=' ' . A No te: AT I. F o r sma ll f re q uency d omes th e differen c e is II Ot rh at si onitrcan t. . ~k.L 12 S ma ll trianqular w indows occu r The cen ters of these "w in dows" are foun d an d are u sed as the vert ices o f <J 3 ·way gri d fo r t he PPT .. Usuallv. The e le m en ts joi ni ng th e t re nsla te d vert ices form t he chords of iI 3 ·w ay ureat c ircle gr id . '\ " .. ~ o n s alonq . A z. 12v hn« 12 etc .lliloOri~ each new II iangl e liIiiIiiidIcI_t 'n It. -: Of I c 3 A (a ao) ~ [The c hord fac tors en d other da m we p Uh li" h" d 1".= a-I s. Throu gh ro tat io ns an d refl e ct io ns o f th e bas ic u n it and its subdivi si o ns t he e nti re 3·way gfldd mg o f t he PPT may b e fo und .. I Z A - _ ~V I A ~ £j I 13 Nof&: Ihi s me th od is somet im e s re fe r re d to as the requla r t riacon ta he d ra l ue ode sic grid a nd WaS d e vel oped b y Du ncan Stua r t. 1. 0) .] kf~t./u. .l . 0. s m a ll u ianqula r " w ind ows" occu r in th e gr id ." Ril l!1 to 12 Ai'.«. ". 'jth parts chosen as e oua l arc d ivi sio ns of th e cen t ral ang le o f t he po ly he dr on.-.. Noie : a.. .. alterati on (. No te : ABC is a r ight tr ia ngl e Aa. 0) o f th e po lvn eor o n. fre1uenc:r mere tlses yeomefrtc<j!y h) th e c ho rd cf ~ fa c to rs m a y be Calcula ted b y th e A/ A ""--= " '-I I "J I I I I I I 2 (sin 6 /2 where cf c hor d f ac t or J ' cent ml an gle < I >-J c:. ~ A Note : 11 AB. 0.L U r Not e : A 11 1 2 (oo. ?. Th e rus t o f ttre procedur e IS th « same.'. The PPT ma y h e desc r ibe d as si x righ t trian gles e ac h be ing a re flec ti on o r rot a tion of t he ot her. are pr o vid e d t h rough t he perpendic ular to onv give n polar axis ...• 6 v hDS 6 a i ttere n: st ru t len g t hs.11.'p!?lI1e. The ve r tices are t h en t ra nslat ed on to t he surfac e of the ci rcu mscribe d sph er e alon q a line passing t h rou gh t he resp e c tiv e ve rte x an d the o rigin (0.1f year un d er t he na me "Triocon" w er e d eveloped fr om t h is m et ho d.. Th e.1planes.:-v .12 fy i. -t :)h Wlrldc m are etllil liltor'll l \r ~~ nyi arts c hose n a ..O)~~r. f ~ r. sma ll equila tera l tria nqut ar r In the griel. DOa td th e method o f subd ivision . ( ' I i j A The ver t ices of t he 3 ·wa y gr id are t he n t ransl a ted to th e surface o f th e c irc u mscr ibe d ph ~ re alonq a line piJssin g t h ro ug h th e resp ect ive vertex a nd th e o r igin (0 ...\ '::"J..0) metimes referred t o as the a lte rn a te geodes ic gri d .»: ( 0. B v has 8. fall it' y In the equ ato rial region ...a ~ . Me t ho d 2: T il e pr T is su b divid ed in to nl rnq uencv w it h th rho c( !n tl ~ 1 "nu l ~ (J f l h l! polvnpdru n.\! (~ \' " :> '\ " .waY gri d co nnec t alt e rna te po ints o f su bd ivis io n of side AS to th e po int s o f d ivisi o n of side BC.i..O... IJW. rtote: .~ . How ever. } en d qeo me tricettv w ith the eltemote [Clnss I)." previous . .. . A P J./ grid. _~"----"' A >l b) W · I'J No te : A B is p. .gaJ2 ) "-' a ll an \lle fo r me d by an e lemen t an d a rad ius from the ce nte r of th e polyhed ron mee ti ng in a com mon poi nt... Th e e leme n t con necting t he translated vertices form t he ch ords o f a 3·way grea t circul a r grid . \' \' \' " .at ea ch end of an ele me nt.JS "Ciuss I ".. ItW Mt we gcnC'(dllv colt "st terns tc" b reakdo wn in this b o o k t en d i n Dorneb ook Orvc) ...- tJok. Cl in to n r... f~ of the dfhedrM Fac e angl e ( a lp ha 0< } = an ang le fo rm ed by tw o e le ments mee t ing in a comm on point and ly ing in a p lane th at is o ne of th e f aces of th e polyh ed ron. a rnnOIEI-e \.. Three orie nta t io ns a re co nsi de red: .. Jo e work ed w it h Fuller on his progra m s and was fu nded bv NASA on a pro iect ca lled "S truc tu re! D.--Jl .base d on re gu la r po lyh ed ra l fo r ms ..- " . . wh er e: 11 = )fjy'2. xplo ring ' and ex pan ding t his field .' AI p /~17e 4 =' 1"2AB C I S . T he verte x o f t he axia: an gle is c hosen as that poin t c ommon to t he polyh ed ro n e lement a nd rad ius. 0. an gles etc ..d 11. a nd th e e leme n t is th e co m mo n line. ... -z:- V ~ no._. &. 1\ Pr incipl e side (PS) = a nyo ne o f th e sides of th e pri nc ip le pol yh edrol t riangl e.(f..tf ffie.. fJ..8 ) '" a n angl e formed by two pl anes meet ing in a co m mon line . ~ ico sahedr on edge b is kn o wn th e c ho rd fa ct o r may be cal culated as follow s: cf = 2 si n~ 2 Th e length of any e le me n t for la rger struc t u res may be found by t he eq ua t io n : 1= cf x r Class II : ...0 _ ~ ~-~/fl. Th e ve rtex of th e ce nt ra l an gle is c hose n as that point co m mo n to both radii (the ce n ter of the pol yhedron ). using the d itterem' m ethods...".. Centra l a ngle [delta b ) ~ an an gle formed by tw o rad ii o f t he po ly hedro n passin g t h rou gh the end poi nt s of an el e men t o f the polyh ed ron. computer .fre q ue nc y o f subdivision ma y o nly be eve n . To me asure th e di hedral a ng le m e asure th e a ngle wh os e vert e x is o n the e le me n t of the dihedral an gle and whose sides are perpendicular to th e element a nd lie on e in each face o f t h e dihedral angle. . Th e gen era l p roced u r e was co find tho 3 -di m ensio nal coor d ina te s of th e ve rtices of the grid On the sphericet su rface .freq ue ncy of su b d ivis io n may be o dd o r even . o f the pol yhedr on is used for ca lcu lat ing th e geometrica l pr operti es o f the st r uct u ra l c onfigurati on. :. Pil l ow D ome. can IIOt sotis f y th e range o f con di tio ns t ha t I S! be geome tr ica lly and st ruc tu rally met. ill its fund ame ntal st a te. Th e rem aining fac es m ay be found by rotati o ns an d /o r ref lec t io ns of th is princ iple p olyh edral tr ian gle and its t ra nsfo r ma tio ns . Ch ord factor (cf ) > the ele ment len gth s ca lcu late d ba sed o n a rad iu s o f a non -d im ensi on al unit of o ne for th e sphe rical fo rm with the e nd p oints o f t he ele me n ts coi nci de nt with th e su rfa ce o f th e sphere .) whe re: V = no. Wirh k no w n pa ram eter s and sop hist ica te d ana lvs is.0) fz . of verti ce S} F = no.2 E .~ ~l "\ Math Jose ph D.. Fo r ex ample th e Expo dome in M ontr eal is a com bination o f a 3 2-freq uencv regul ar trie con (Class II. Fr equency I Nu Y su b di v ide d . 0) of th e polyh edro n a nd it s res pec tive vertex . The tech n ique h e uses in vo l ves an a ly tim l geo m etry (Fu ller used sp he ric e l trigon om etry ) and th e cal culations are d one w ith . GEODESIC GRIDS Up o n using t he sphe rica l fo rrn as a str uct ure l u n it . (0 a 0) --'-~t-~ :axltL/4. o r po rt io ns o f o ne fac e. The tw o plan e s th em selves a re faces o f t he dihed ra l a ngle. ( A • .. Th ere have been ma ny meth o d s dev e loped fo r reduc in g t he ba sic po lyh ed ra l form int o 0 la rger nu m ber of componen ts f rom wh ich the geo metrica l pr operti e s may be m ade to remain w it h in t he struct u ral fa bri ca t io n and erectio n Iimi ts for a d esir ed confi gu ra ti on .£ 2 cent(}y.. / Th e m et hod s d esc r ibed her e m ay be cons ide red as hav ing cha racte rist ics of o ne of th e two foll owing class ific atio ns: 1!" /. Alumi num Su n D ome. with the pa rts ch osen t he three princip le side s. .re spect ive sides th er eb y giving a 3 -wa y gr id so th at a ser iex o f equ ila te ra l tri ang les are form ed. . D o tu eb o o k comments arc in i talics . F . an d th en to calc u la te the ch o rd l engths. V =iL+2. Several m et ho d s of qen erat inq 3 -way geode sic grids are d iscussed he re in a bro ad se nse to give t he e xperi me n tel' a hasi s from w hi c h ot her meth od s m ay be devel op ed. 'OvL 6~ 180 - (1l.-. (0. me t ho d 3) an d a 16 ·fr equen cy tr un cstshte alternate (Class I. Eac h point o f SUbd ivisio n is t he n co nn ec te d with a line segme n t parall el to th e i. The ax ia l a ngle. CLASS I Method 1 : { This me t hod is wh at w as publi. of faces for tota l Ico s~ he dr a l spher e ~. Ori entation = th e ori entat ion th e po ly he dr a l form ha s in space w it h respect to t he o bse rve r. . . (o 0. what w e c. Th e ce nt ral angle Class I: .. m os t generally the rh o mbi c triacon tah ad ron ."':'. The following sect io n is an or ticto by Jo e Clinton o n the different m ethods of producing geo d esics from the ic osah ed ron . it is ap pa ren t that th e basic po ly hedra l T'U f o r m .20. m ettiod 3). If the ce n t ral an gle ~ o)' y.~' . o{he ral ()f f...... o f e d ges V = frequ ency o f SUbdiv ision figu re'._ . . Du e to t he sy m me tr ica l c haracte ristics o f th e b asic polyhedral fo rm o nly one face .. J O f1 classifies . .'~ drhedral icos a hedron edge & Face = any of the pla ne po lygo ns mak in g LIp the surface of th e struc tu ra l fo rm.~sign Co ncep ts for Fut ure Space M i ssions " . w r ote this sec tion md i fl lv witt: the in ten t o f corn rnunicetinq the state o f deveto omen r of geodesic qc o me tries an d th e hope ttuu it w ould b e an ai d to thos e i n tereste d in ."v . . Th e sp ecific motivation for d evel oping th ese m ethods was to ha ve a variety of fo rms to comb ine in large sp ace f ram e d ome s. . o f face s for t otal rhombi c tr iac ontah edral spher e E '" no . BASIC DEFIN1TlONS Axial an gle (o mp.. Jo e.) wh er e: r ~ th e rad iu s o f the desired struc t u ra l f orm I = th e length o f the new e le me nt Dih edr al an gle (b eta...de m OllSt rat es sy m met ries as illustr at ed in the followin g e /ertff'l-r/. largo str uc t ures can b« op ti mized IJv a ittsr en t co m hin n tions an d differen t me tho ds.... c A fO IT'l e qua l d ivisions a lo n g --" p(l/rtJecJrttl TrI"Mjleslide- ?nnap/e.. trowever . I = the number o f parts or segme nts in to whi ch a pri nc ip le side is Note : AB is par all el to 12 n E'. he clessltlc s as " Class 1/" ... edge face a nd ve rt e x.. mo st gen er al ly th e icosah e dro n.- f:jt/1/:rfa70-1t:>.a.. F = 2 (it) E = 3 (fl.il . and is th e geometry of th e Pacific Dame.. ..based on th e qu asi-regular p o ly he d ra l form s. METHODS OF GENERATING g-WA .'JI' " trieco n ". m~I d~ x MAA AA1A zr ?v 4-V ZV qv (£jl' 10il Each vert e x o n th e PPT is then t rans lated l. of verti ce s } F = no . Princ iple p o lyh edra l tr iangl e (PPT ) = an y one of t he plane equ ilate ral tri an gle s wh ich th e fac es of th e reg u lar polyhedr o n. E '" no. . V "' IOV"t~2 b ma y be found by kn o wing th e ax ial angles a .~f pO I"hed/'7lrr ... on to t he su rface of the cir cu m sc ri bed sphe re . with these co or dine tes and an al ytical torrnutss . f or sma ll st r uc t ur es (up to 4 0') tnov are not gen erallV t elcvent. of edges V frequ ency of su b div ision / .c. .. K ~ _. e tc.0. .] Cl S Th e PPT is su b d ivide d int o n fr equ ency .\he d i ll D o rn ebo ok One u nder th e na me " alternate " ..~ " .(lma y be f ound if t he ce ntra l an gle ~ is known by t he fol low ing equati o n : = 180.. a lo ng a lin e pa ssing t hrou gh t he o rigin (0. Face angl es refer t o angles of tri angles generate d by cho rd fac t or s. Since t hat time . regardl ess o f what port io n o f it is u sed . just as the fac e in th e origi nal basi c icosah ed ron wa s an equilateral triangle. t he 2-frequency fac e 0 .0) and vert ex LO. It's easy to rnake a mistake here . 2). w e a re pr intin g Class II. On p.1 at one end and vertex 1. and again in the morning. th e math emat ics we have u sed for most of o ur d ome s. because that area represents one icosa isce brok e n into two parts at th e edg e .0 2.1 1.1 (Fig. the number is given in degr ees. Th e foll ow ing is an explana t io n o f th e tabl es . an d so me 12 t et rah edron and oc t ahed ro n cho rd fact ors. The anqle given wh o se apex is at th e second vertex identi fic at io n shown . . O n t he pa ge o ppos ite we are pri nting Cla ss I. 110.r-----"':'. I. 3). it w ill be III th e same unit of measure as you measured the radius. Note th at the diagram . multiply th is chord factor by the radius of the dome you want t o build. It y ou want to ex plo re ful ly the d iff e rent meth ods we sugg es t t hat you mak e mo dels a nd st u d y Clinton 's geodesic m athem ati cs sec tio n . as at EXP O '67.0 1\ c ho rd fac tor is a pur e nu m be r whi ch. d iagra m lor Class I i"a lte "na t" '" brca kdo wn .2 is also equilateral and thus symmetric. 1 ico sa fa ce sho w ong f-A CE ( 0( A XIAL I a: I. edges.0 1. the strut length will be qive n ill inches.frequen cy Class I (" al ternate ") brea kdo wn FA CE a ngle rtesc r ibed by 1. "Advanced Structural Design Concep ts For Future Space Missions. and is called that in the rest of this book . 0.0 cen te r of sph ere incl uded an gle b et we en faces sh aring 1. Th e triangles of the same shad ing ar e t he sam e in all respects ex ce pt left and righ t or ientation. It is t he total included angle and agai n is given in d egr ees . F igur e s o n the o ppos ite page ar e for icosahed ro n based Class I geo d esic spheres as dis cov er ed by R.0 refe rs to t he angle between the two faces th at sh are edge 1. We o btained th e c ho rd fac tor s we pri n ted in Oom ebook One fro m vario u s sou rce s. a s described by Clin to n u n pp .0 2. in th is case . Th e numbe rs are from c o m pu t er readout generated by proqrarns deve loped by Joseph O. not spherical angles on a tru e sph ere. It is the angle that a strut meets its hu b from the side view. 3-. wh e n multipli ed by a radi us. and iill exam p les give n ar e fo r a "tw o fre qu ency " breakdown . FACE 1.0 (Fig. gives a str ut length. Big dom es u sually are mad e fr om folded p la te s whi ch giv e the sk in a ~a r ge c ross sec t ion. It will be this number minus the diameter of a hub. This length number is also known as the CHORD FACTOR. The figures V(O) : 42.0 1. 6·. An ic o sa hed ro n ha s t wen t y ide n tical eq u ila tera l triangle fac es and twelv e pentaqo nal vertic es. we use on ly that p a rt part of the diagram o u t line d by 0. Aga in.0 V I L}~6 me an s that the number of 'vertices in o ne ic osa fa ce of thi s par ticular br eakdown is 6. Th e n umbers on th e d iagram are vertex iden ti fications. Starting wi th th e two fr equency sect ion: 2·FREQUENCY ICOSAHEDRON mean s a 2-frequency br eakdown of a ba sic icusa. but only to the line from t h e hub to the sphere ce nt e r."=------"'::'. ve r t BX idtmt if ica t ion Figures given do not refer to structural strength. an d are ref err ed to in th e tables start in g w ith 0.0 This means we are talking a bo u t the st r u t that has vertex 1. 1. Note that the pattern of sameness is symmetric about the center of the triangular array o f whatev er breakdown frequen cy you are using. th e mor e sp here-like th e ico sa hedron becom es.1 1. Th e highe r t he fr equ ency. Lo oking at Fig. The d iagra m h ere is on e face o f the bas ic ico sah ed ron (d ivide d into t he n um be r o f additi o nal faces as requ ired by th e des ired freq u e nc y. 1970. this num be r will include the hubs. if you give the radius in inches.1 0. Figs 6 an d 7 sho w the 3 and 4 frequen cy breakdowns . F(G) ~ 80 ar e the number o f vertices.0 refers to one angl e of the triangle des cr ibed by these po ints . E(L}: 9 m eans that the number of edges in one icosa fa ce is 9 .0 1. This length is vertex to vertex.0 1.0 • • • . '0 . so the actual cut length of th« struts will be less than this number. m etho d 1"altern a te". 4 . o r th ey a re m ad e fro m tw o domes o f the sam e or diffe re nt frequ enci es bu t d iff er en t size . 5 are e xac t ly the same . E(G) : 120. . ) Th e d ifferent le ngt hs o f ed ges of the se add itio na ! faces cause the ico sa face t o assum e a more sp h erica l shap e . 2. and the strut 1.0 1.1 st rut 2. met hod 2. Th e variou s " p aragrap h s" in th e tabl es will y ield all the ne cessary info rm at io n if you keep in rnind that. Note that it is not the entire included angle under a hub. we h ave made coni act w it h J oseph D.0. 9-freq uen c. Thi s m e ans that th e three tri a ngle s shaded in Fig.ies (multiples of three ) have a point at the cente r o f the face ins t ead of a triangle as in thi s case. On p.0 AX IA L ang le /~ I I I I D IHED RA L an qte is to tal • LENGTH 1.2 e- ~--------. The result w ill be the length of that parti cular strut in the size dome you de sir e. Think it out ca refu lly." F inal Report. Buckminster Ful ler. It also means that th e ce n ter tr iangle is eq u ila ter a l. C1in t o n wh o w orked o ut th e glJodesi c c o mpu ter pr ogr am s fo r NASA. AXIAL 0.1 1. 106·107. Wh at we call "tr iaco n" is CLASS II. NASA Contr ac t NGR 14 -008-002.uld 01 HEDRAL (f3 ) a ngles.-~--------=rr---------. If you are using hubs.0 2 .0 1. In other words . Wh nt we ca lled "a lte rnate" bre ak dow n in Domeboo k One. a nd in mu ch uf thi s b ook is cl ass ified in the computer pr oqr arn s as C LAS S I. o ne ins id e th e o ther and laced toge t her w ith addi t io na l str uts . To use it. t he f la tter tn e a ngle of th e do me's fa ces (th e " d ihedra l" below ) and the more cri t ical Is accu rat e w orkm a nsh ip to prev e nt "popp ing in" o f a vertex u nde r load. Clinton under a NASA -sponsored rese a rc h grant. =-------Y 6.0 2. l. t he m ath w e hav e used in o u r "rriacon" dom es. 90° minus the axial angl e equals an gle you cu t or bend struts . 1 we are p rin ti ng so me alte rna tive meth o d s of both c lasses .1 refers to the angle between a line drawn from the center of the sphere (0. The value is DEGREES. Draqrnm is one [ace o f icos ahe dron . th e h ighe r the frequ en cy .2. In eac h class th er e are seve ral methods.1 1. does not imply size. 2. and faces in an entire sphere based o n th e 2-fr equency breakdown of the basi c ico sa.0. we see that for a 2-frequency breakdown.1 0 .0 at the other. 01 HEDRAL 1. F(L} :4 me ans that ther e are 4 faces in one icos a face wh en brok e n into 2·frequ ency . See Fig . March. For a given d iamete r. 1 4.' .1 2.179636 60 . 4 V( G) = 42 E(GI ~ 120 F (G) 1.807496 5.0 c 1 das" I IIl1'thud 1 12 E(G ) . I ~ ~ . 1 2.1 1.1 1.856775 0 .1 4.1 6.· 6 E(L) = 9 F( Ll .96 11 15 A X IA L 0.1 1. 1 6.0 0. F HEl) lIE1\CY ICO SA" EI)HOl~ class 1 method I V (L ) ~ 28 E( L) = 63 F(L l ~ 36 .0 4.19 441 6 0.058991 59. 1 0 .-.1 2.2 4 .1 1.1 1. 1 4.2 4 .1 LtN GTH 83.170748 5.0 3.1 3.099906 60.318399 LEN GTH A X IAL FACE FACE and fe rent . 1 0.54653306 74.470501 173 . : . 5.0 0.190476 86 84.29858813 8 1.534353 59 .0 ~ 80 LENGTH E AXIAL FACE FACE DIHEDRAL 3.1 5.30 F( Ll ~ 16 V(G) .34861548 79.2 0 .0 0 .1 3.0 84 .79 7728 FA CE 1.5831 64 165.0 58.1 4.730537 2.D99996 63 .0 59.1 3.0 60 .7 17473 169.0 5.0 71.61803 399 71.0 2.0 3.0 0 .708 4 16 166. f' ace s Ihe s".7301 90 FACE DIH ED RA L 4.634727 165.0 F(G ) = 720 0 .0 3.3 6.541075 4.1 3.0 1. j . 7.2 82 . sh"de<1 fa ces Fi r:.0 2.565048 169. 1 5.0 3.0 2.2 6.6687 68 58 . 3.1 4.838518 Fac ('!s t he same co lor are id e nt ica l except that wh ite f ac es o c cu r i n p Hi r s as " left s" ornes.2 FACE 62 .0 2.1 5.1 3.19801258 84 .2 3.1 3.1 4.1.1 171.0 2.1 0.65029 2 59 .2 4. 948445 57. FA CE 4.2 ~ 1080 1.2 DIHE DRAL \ ~ LE NG TH A XIAL FA CE DIHE DRAL 6.1 4·f n~qLJ enr. .0 1.1 5.2 6.1 2.0 1.2 5.0 2.0 6 .1 2.1 0 .1 78 .430325 LENGTH AX IA L FA CE FAC E 0 .3 0.0 1...54 2280 I .0 6.3 5.359272 58 .0 6.26 159810 LENGTH 3.1 2.0 2. I I £1f~1\ . f f 171.2 3. \ lIE () I W l\ E( L) ~ 3 F (Ll 1.981901 0.623971 57 .178770 0..0 5.818582 63 .9 99996 LEN GTH A 68. 1 1. 5109 21 1.0 5.2 3 .2 4.2 5.0 3..1 1.2 4.0 1.. .1 2. 7 I I 5.1 58 .0 '\( \ L I * F i/!.2 6.350295 58.86 1974 AX IAL 59 . 1 3. and I".2 3.i-I'HEOlJENU " I C O ~ ."""'" .1 3. 1 2. (j 'Y' .3 1286893 80.1 0.0 5. 1 0.NCr Ir:OS \IlI'.1 2.0 2.2 3.16271 2 A XIAL 4.0 5.324919 69 8 0.1 4.090703 6 1.0 4 .ml I V( L ) . ••• • .1 1.78 1497 55.M' ~ iIifI!" I aLL .1 2.544544 DIHEDRAL 3.2 1535373 83 .0 2.1 2501 7 4.0 3.1 L ENGT H A AX IA L FA CE FA CE DIHEDRAL LE NGTH B A XIAL FA CE DIHEDRAL 1.0 I 4. •• F IU: Q I I~:'~C" ICOSAIIEDHOi\ el:l ~ 1 nu.2 5.2 3.0 .843306 0.0 4 .1 3.807500 3.-1 .2 59 .25510701 4.6 17161 0.090701 58.3 5.0 O.0 1.1 3.3 6.2 0.611144 63 .2 4.0 1.1 3.25318459 8 2.2 4.0 2.244049 174 .1 5.~"" .1 92 E( G) LENGTH 8 AX IA L FA CE FACE DIHEDRAL LEN GTH C A XIAL FACE FA CE DIHE D RAL LENG T H A A XIAL FA CE DIHEDRAL LE NGTH 8 AXIAL FA CE DIHEDRAL 1.569 0 10 157 . 1 0 .0 2. 1 4.0 2.31804 6 57.43 2868 .9 61 621 54.0 1.0 3.0 2.0 58.-H . 1 58.FREQ UENCY ICOSAlIEIlI{():.1 2.1 0 . .7272 77 54.1 FACE 59 .1 3. Bc for e .0 71.0 4. 2 5.3 0.1 0 .0 1. 50 573 7 1.337645 54.0 4.0 5. 1 1.1 4 . 2 6. L .1 5.0 65. 2 60 .0 1. \ \ ( I A .231 79025 1.1 2.99999 7 FA CE 3.0 3.1 4.~ 2.0 0 .131815 3.2 6.4 139 79 58 .0 1.0 4.178778 0.0 0.750 F (G ) -r 500 L ENGTH AX IAL FACE FA CE DIHE D RAL LENGTH AXIAL FACE FACE DIHEDRAL L EN GTH AX IA L FA CE FACE DIHE D RAL L ENGT H AXIAL FA CE FACE DIHE DRAL LEN GTH A XI AL FA CE FA CE DIH ED RA L LENGTH AXIAL FACE FA CE DIHE DRA L L ENGT H AX IA L FA CE DIHEDRA L LENGTH AX IAL FA CE DIHEDRAL L EN GT H AXIA L FA CE DIH ED RA L (I.1 0.1 1. ne c o lo r c ue identi cal \ / I " '~i: '~ .0 5.0 2.1 5.1 174.1 3.6700 21 LEN GTH 0.2 5.3 44 74 1 A X IAL 0.765753 83 .1 1.2 0.1 5.1 4 .1 3.-- and "ri gh ts. 1 2.0 2.0 0.1011 35 4.30 F( G) 1.~ ~ :1.1 0 .18190825 84 .3 5.29524181 1.1 4 .1 2.64208 2 0. 1 4.1 0.1 2.O 1. 1 0.1 2.240530 0.1 4.1 4.0 4.0 2.9708 92 FA CE D IHE DRAL LEN GTH f AX IA L FACE D IH ED RA L 1.216 628 21 83.61114 3 173 .2 4.0 1.1 172.0 4. 1 5.E5i .0 AXIAL 56.137622 173.1 2.189684 1.1 6.2 0.1 1\..490046 0 .42 144 2 0. 1 6. 1 2..2 0 .1 60 .0 3.0 3.0 2.3 82 .159 766 172.0 0.0 1.0 5.989368 173.1 41 • V(G) 362 E(G ) 0. .1 2. V (G) ~ I-I·H EVI F.0 2.41 231 9 0 .1 0. 2 4.2 5.1 5.5 64739 0.33 160 4 1.197807 FACE 2.1873 8340 84 .0 0 .41 24 1149 78 .35 9272 0..24508 578 LEN GT H 3. 534954 71.1 0.0 1..53 15 10 57.141260 55.2 82 .999 99 9 62 .20590774 84 .4035482 1 1.444659 FA CE 2.1 3. 641064 2.. due i lk .0 70.thorf t V(L) V(G l ~ 10 ElL) ·.1 4.1 FA CE 6 1.2 945 3084 8 1.0 desiq ncd ( ears ago.'!t.1 6.0 4.429 183 173 .1 3.0 3.1 59 .3 5.0 2.1 171 .2 171 .920114 169 . ~ II urm mT0I151 em RrH.0 4. " A~~. 2 0 .2 3.0 60 .1 2.0 1.0 2.1 83.1 4.1 4.1 1.0 3.244049 173 .0 4.20 28 1970 84 . 1 D.1 4.\ I1lW llO \ (' I ll ~ I IIlf'l hod I V(Ll '.0 5.1 3.0 3.0 3.0 1.0 2.0 0..0 58.0 2.0 4.334194 169. 1 81..2 5. "18 185 7 3. 186995 D IH ED RA L 1.0 2. ~.1 3.!.2 3. ~ .05 898 9 173 . 2 5. 948447 172. le d .0 1.23 '159760 I DIH ED RA L LENGTH AXIAL FACE FA CE DIHEDRAL 0 . 1 2.1 = 20 V I L.898845 A XIAL 0.1 1. 2 \ .1 5.0 3. /\ .9535 90 5. 1 5.7 839 24 0.- ~ .477804 D I HEDR AL 2.1 1.3 0.1 3.0 4.0 2. 1 0.1 5..0 61.1 2.1 3.1 3.583 164 1. ? . 1977 90 2. ~~ .403753 5.0 4 .2 6. 14 1629 175 .2 1535 373 83.553503 DIHEDRAL 3. "~ -- -"- .2 5.1 3.0 3.484 227 A XIA L 0. 1 3.1 3. 1 82 .y .1 2.» >: 1\7' // V <\ I I-FIlEQl a ::'\c:Y ICOS.0 5.28 25 25 1. -.1 6.1 5.- -~.36 93 23 FACE 3.0 0.20590773 84 .0 2.0 3.0 3.1 82.99 9098 16 1.2 3. 1 4.2 3.2 0 . 1 0.1 3.964814 FA CE 3.1981471\3 5.1 5..204537 17 1.17 9635 66. 252 E(G) -r.0 3.2 4.3 5.0 3..24534642 5.. 1 4.1 78 .1 3.1 3.7084 16 60 . 18 ~ F{l) · 9 O' \ B /\ 7\ C A ." f'i.1 4.22568578 4.1 5.999998 138.2 4. 1 3.\ class 1 met hod ) V (Ll ..0 173 .0 5. ' ~~ 5.1 4.2 6.0 2. 1 5.0 270 F(G) 180 1. 671\566 FACE 3.0 4.1 0.0 3.2 5. .2 3.796873 169.0 1.72 1\745 1.0 5.2 4.0 5. 1944'L7 172 .0 0.15 E (Ll .2 5.3 6. 606874 2.5 90 8 18 1.2 5.U 5.520774 LE NGTH 0 .2 5.0 1..070368 171.0 3.0 3. 1 5. .~I LENGTH A X IAL FACE DIHEDRAL 6.1 4 . .2 6...0 4.2 5. 1.0 63.0 1.) ~ 2 1 E( L) " 45 F Ill ~ 25 V( G) '. 0 11 Ihi ~ pagf! an' Churd Faclur~ fur: II d:ls~ ) [Al terna to) method) 3.2 4.40354 822 3·f rL'q ueflcv .1 4.' 2·frcquency . I LENGTH A XIAL FACE DIHEDRA L 6.1 6.1 0.0 3.0 6.0 2..0 4.1 3.3141 62 54 .470502 61 .202989 0.1 0 .0 59 . 7 liNl.2 3.0 3.J ".IJHON C ll•• i I IIu:ll.745039 0 1HEDRAL 4 . 1 3.162 E(G) ~ 48 0 F IG ) ~ 320 L ENG TH B A XIAL FA CE FA CE DIH EDR A L L ENGTH D AXIAL FACE FA CE DIHE DRA L LENGTH C AX IA L FA CE FA CE DIHEDRAL .~ .1 1.818583 59.2 0.3 5.0 6. 2 0.162567 22 85.0 3.1 2.0 3.202995 0 .2549 13 0.0 168.1 1.0 2.05 146272 1.3 6.C LEN GTH A XIAL FACE DIHEDRAL 1. 1 3. 1 1.1 3.247 24 291 LENGTH 2.0 2.0 1.1 1. 1 4.0 3.2 171 .0 59.2 3.20 281 969 84. 0 2.5 381 .2 3. and to all the chord factors and angles ill the book: The tabl es f or Class II fol low the same f or mat as tho se f or Class I excep t th at th e verti ces are ident ifie d as in th e diagram : Th e foll ow ing compar ison between Class I (Alternat e] and Class r I (T ri acon) breakdown is r epr int ed fr om Edw ard Popko' s Geodesics.0 0.0 2.36284 80.0 1. 1 3.0 3.409 1 .0 1.0 1.~ LENGTH-lO AX I A L AX IA L 3.170 2t"i 388 0 .0 2.2459 92 81 .0 1.2 3.2 meth od 3 2.32840 70.1 1. AX IA L L EN GTH·1 5 AX IA L AX IA L 4.26700 170.0 0.1 3.0 3.206 04 .7 895 9 161 .1 3.77250 80.1 2.1687 1032 85 0 7' 0" 84° 16' 44 " 850 12' 36 " 84°23'1 1" 85° 22' 58" 84° 41 ' 40" 85° 36' 11" 85° 9' 41" 1.8740 .0 2.0 0.0 3.1 3.71 3644 71. No co m p lete gr ea t ci rcle delineat ed na t u rally by t he s truc t u ra l paner n-vsu ch as t he eq uatoria l ob ta ina b le o n th e eve n nu mber Alte rna te breakdown s. Meth od 3.0 1. 1970 NASA Contract NGR 14-008-002 The general instructions 011 p.0 2.1 2.1 1.0 4.09484 67..26700 54.72004 82.190"703 "79..1 3.0 2.0 3. In even freq uen cies .3 76 7 79.08 695 83 .0 2. i 2.1 2. A symmetry of re latio ns h ip 0 1 adjacen t f aces ' ma king eas y co mb ina t io n in to diamonds.2 1.9J2"7 77.2 3.504 34 82 .0 2.65686 83.40696 71.0 1.376 7 . Buckminster Fuller. The f igures are based 011 Class II geodesic spheres as developed by R.0 1. A great er var iati o n o f mem b er len gth th an with th e Altern at e sys te m .0 2.0 0. 1394 .2 1.1 1. an d excp.1 3.22 6708 73.1 2.2 1). 1 2.1 3.0 3.0 1.0 2.1 2. 1286 _331 518 "18.167 016 11\ 3.0 2.82 582 59.0 2.1%(-)3944 0.94600 1.1 7545 166.0 1. The numbers are from the computer readout generated by program s develop ed by Joseph D.1 1.0 0.0 2.0 3.0 3.1 !J.0 1.0 2. 1186 . Co nse q uently t runcate d b ase member s m ust be used in every case .0 2.0 U.2 3.33949 1 79 .0 1.3 2840 56.10 875 6 7.0 0. 1 2.0 3..66866 56.0 0.3 3.0 ~ .1855 '78 79.70 90 80.0 I}I 1.1 0.0 90 20 53. quende.8 7730 170.0 2. 64300 171.2 3. 1 1.0 2.065 69 166.5470 78.0 1.0 2.mgles.0 3. .3 . see di agram rig ht .0 1. March . Th e num ber o f d iffe rent co mponen ts in re lation to fr!::'qu e ncy in cr ease o n a geo me tric scal e .0 1.1 1. In pr act ice.2 2.0 2.1 3.40690 83.1 3.1 3.6 6 16 82.0 2.1 1.0 1.0 3.65000 171.'2.1 2.0 2.0 3.2 3. " A dvanced Structural Design Concepts for Future Space Missions.68770 174 . c. 1 3.24 7825 78.06170 171. A m in imu m vari ati on in m ember le ngt h.27 622 9 82 . 1 2.26 700 71.224 25 . 1 2.U con/tnt. b.0 3 .0 3.1 0.47280 174.1 3.1 3. c.0 2.19937 078 0 .30352 8 80.0 2.0 1.1 3.1 1.581 40 54.1 1.1 2.200 6 2. 54 61 82.0 1. so ach ievin g hemispher es w it ho ut t he need f o r trun cat ed mem bers at th e ba se.0 0 .1 2.1 0.0 0..0 1.0 2.1 3.691 b 75.1 3.3 3.0 0.8638 . 58140 70.0 2.0 3.0 0. 83. a mo re gradu al va riat ion in sca le o f br ea kd own is avai lab le.0 1." Fin al Report.1 2.0 3.0 2.0 2.81977 53.. .0 3.1 1.> 2-Io'REQUENCY ICOSAHEDRON class II method LENGTH -A LENGTH B A XIAL A XIAL FA CE FA CE DIHEDRAL DIH EDRAL 0.0 2.1 . T HE TRIACON BREA KDOWN The triacon br eak d own has th ose advantaaes over the o t he r breakdow ns a .1 2.0 1.2 3.1 2. \ \ \ \ \ 4. 108 app ly to the below cho rd factors end .1 0."19 9 4 79.0 1.3 .0 0. As odd and eve n num ber fre qu e ncies a re bot h both o bta inable.3 3.58971 70.2 4.0 3.2 On Ihispage aro Ch< l Flid o rtl fo r: m . 1 2. 1 3.2 n um be r. CIinton under a NAS A-sponsored research grant.2 2.1 4.0 3.0 1.22425 .0 1.2 3.0 0.16 103 172 0 .246953 74 .0 1.3 1740 78.1 2.0 2.0 1.329 151 79.0 0.0 2. b.0 1.1 3. 1 3.0 1.72004 84 .1 1.333333 82.1 1.0 0.5044 0 83.2 2.1 . 1 2..0 1.1 2.0 0.3 1740 80.0 2.1 2.0 ~ 'V f-- di agr am abo ve s ho ws o ne of th e 0 iden t ica l right tri angles in th e icosa fac e.0 1.46 610 54. l co se fa ce shown b y d otted lin es 0.60280 174.3 3.3 7576 0.\ X I A L. 1 4.1 3.1 3.0 1.16566 153. 1 2. A XIAL AX IA L AX IAL AXI AL AX IAL AXIAL AX IA L 0. Fro m do tted lin e up is one of th e 6 id ent ical right tri a ngles in the ico sa face.1 2.1 1.46610 54.21877 .2 1.1 1.1 0.2 2.0 1.1 0.45830 66.090 20 56.0 2.smus t al w ays run in <I n even 6·FREQUENCY ICOSAHEDRON class]] method 3 LENGTH-A LENGTH-A LENGTH-A LENGTH·B LENGTH-B LENGTH -C LENGTH-C LENGTH·D LENGTH-E LENGTH-F AX IA L A XIAL A XIAL A XIAL A XIAL A XIAL A X IAL A XIAL A XIAL A XIAL FACE FACE FA CE FACE FA CE FA CE FACE FACE FACE FACE FACE FACE FACE FACE FA CE DIH EDR AL DIHEDRAL DIHEDRAL DI HEDRAL DIHEDR AL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DI HEDRAL 0.33609 . Di saovan taq es : a.329 15 1 78.0 1.2 2.0 1.0 L EN GTH · 16 AX IA L AX IA L LEN GT H 17 AX IA L AX IA L L EN GTH -18 AX IA L AX IA L LEN GT H-19 A XIAL AX IA L LEN GTH -20 AX IA L A XI A L LENGTH-21 AXIA L AXIA L LE NGT H-23 AX IAL A XI AL L EN GTH -2 2 A X IA L A X IA L 3. th e adva nta ges o f th e Tri aco n become more e m pha tic in th e highe r f ruqu en cie s-e-usuat tv th is mean s in th e larger d iam et er st ruc t ur es su ch as th o se 100 feet Or mo re.0 1.224 25 .0 3.1 2.0 2.0 1.0 9890 57.46610 53.1 3.0 3.2 2.0 2.0 7.26427 .0 2.0 0. mclbtll1 3 L ENGT H L.0 3.0 1.951 1 0.0 0..1 2. a co nt inuous equ at o r is delinea ted . 54 61 .I ve rtex ide nt if icati o n di agram for Class II (" tri aeon" l brea kdo wn.0 1. 1 1.0 1.27480 71.0 4·FREQ UENCY ICOSAHEDRON class II method 3 LENGTH·A LENGTH·B LENGTH-A LENGTH-C LENGTH -D A XIAL A XI AL A XI AL AX IA L A XI AL FACE FA CE FA CE FAC E FAC E FACE FA CE DIHEDRAL !HEDRAL DI H ED RA L orl-lED RA L D! H'EDRA L 0.0 0.2556 2 .33609 .0 0.1 2.1 0.1 2.0 0.1 1.1 1.64300 174.81977 54 .1 5 53 31\60 2. 1 0.0 3. 1686 .31131 69 .38948 . 2 3. 1 3.2 3. wh at we call "tri acon " in th e rest of th e book.1 4.31\394 4iO 4/1 I \ \ I I I I I I I I I .0 L EN GTH-l 1 AX IA L AX IA L 3.0 3.0 3.0 0.1 1.0 1.2 0.26427 .2 4.0 2.1 2. 1 1.1 0. I :2 \ I 'I I bY 3/2 I . 985 60 79.877 30 3.308 46 171.0 3. 1 2.82 55 80. 7485 . St ruts show n in thi s rig ht t riangle are re pate d in t he o t he r five right tr lanpt es o f th e i co~ l ace .848 1 79.3 .0 1.I I i Below are chord fact or s f or Class II .303528 79. r-rp.0 2.1 2.1 1.0 2.3 1 . less var iati on In p face an gles.2 We don't have face and di hedral angles f or 8 fr equency . 3.1 4.0 0.0 1. 1 0.0 1 . 2 3.1 1.1 3.1 81\9 354'1 0 .0 0.2 4.3 1337 .0 1.2 2.1 2 .0 1.1 2. A m inimu m num ber o f d iff eren t co m po nents h. THE A L T ER N A TE 8RE AKDOWN The alternate brea kd own has th e fo llo wi ng adv an tages : a. an d t he follo w ing di sad vant ages : a.0 1.22 4244 73.90110 170 .1 1. EN GTH L ENGTH L ENGT H L [ NGT H LE NGT H LENGTH LENGTH A XIAl.1 0 .0 1. 2185 77.0 2.0 4 .0 4.0 3. 7090 ..-t herc t o re t here is less gra d uation in sca le th an is ava ila b le wi th the Alt ern at e breakdo wn .54750 70.0 0.tU frtJl1/ en~/tpsl-*.FREQ UENCY ICOSAHEn RON cla~ II .flAM II [Tri acou) 2.2 3.0 2.2 1877 .231 H2 83.0 0 .1 1.0 1.5798 82 .0 1.1 LEN GTH ·12 AX IA L AX I A L AX IA L TR IAN GL E 2 L EN GT H-13 AX IA L AXIA L L EN GTH ·14 .2 2. t fo r the pe nt joi nt s.04826 172.0 ~o 1.1 '1.58971 54.0 2.1 0640851 0.50440 82.0 3.3502 73.07843 167. 6147 .494368 82 .l d l / of tho se t ri " ' l ylt'3.2827 . {J--c.. 1.294 8 8 3 80 .0514 80 .0 . 0 2.7067 .77 84 58 73 .1 2.0 1.n se p ar .8362 7:l .1 rt h I . AXIAL T"l II'N GL E 2 L EN (Jn l. w rit e and I ' ll send th em lor compute r ca sts.0 0. as 2 .1 2. 15 28 .450817 79 ..544 692 78 .6860 EGG 2.0 1.l 1 AXIAL A X IA L LEN GT H-1 2 AXIA L AXI AL LE NGTH·13 AXIAL t\X llI L L EN GTH-13 A XIA L AX IAL LENGTH· 14 AX IA L A X IA L avis ivls.57 9 72 3 72 .0 1. 2 .0 1.98 1 1 .4 0 354 8 72.2628 2.30 5 1 75 .1 2. .284 68 2 84 .7 568 8 3 51 5 2 . 1 3.0000 72 . 1 1.0 J 1.B A XIA L A XIA L L EN GTH -C A XIA L A XIA L L EN GTH · D A XIA L AXIA L T R IANGLE 2 LENGTH-E AXI A L AXIA L L EN GT H.2 2. 6 24 3 .2827 80 . t st ru t Ie rrqt h in th ese two tria ngles.0 0.2092 ..2 35 9 82 . 8'. 1 2.1842 . 1 ~1 2.0 3.1 st 40 new 1.3 13561 82 . 1 1.207 2 82 .49436 8 82 . 0 1.1293 782346 .2 Jm es.1 3 .0 1.458 2 77 7 13 5 55 5 58 3 73 .171 3 70 .3( y.0800 80 .9665 .339 0 72 82 .2 on f ig. 3841 .0 ~2 ~1 ~2 ~1 3.2 2.6841 73.0 2.0 .0 1.>.1 2.2 Z. ~ mP vvov <'\ fW rJ. /l car rain 11 chords in II>!! [es t row. 1 1. 1 2. LENGT H-8 A XIAL AXIA L LENGTH ·C AXI A L AXIAL LE N GT H.1 2.1 1. .420724 8 1. 1 2.7840 76 .0 1. 1 1. 1.0 .0 3. o r truouencv . 1 1.4 2 12 74 75 .6155 26 81 .1 .4404 69.0 0.498 661 80.455 3 82 .2 3.7785 81.1 On th is pag" arc Chord Fa d or ~ for: Elliplieal Domes se s 1.1 1.485463 75.5488 1.0 me " barren .84 9 7 .8739 8 2.4 0 354 8 78 .2050 79. 3 178 .498249 8 1.0 2. whi ch en d of t he st r ut It ld moll! ref er s to .0 1.0864 81 .4553 .30 51 . 1 1.9075 ZAF U EGG .1 2 ..5989 78 .o~ .1 2 .4305 .788 84 9 78 . Lenot h d31 c on be lJ ~ !'I:I ill I he '.1 2.78 884 9 79 .8 739 . 1 1. 359 09 4 79 .1 4.6929 .1 an~ 1.2740 ZAFU 1.2 125 .0 6 8 7 75 .56 2 77 3 7 6.1 .4 68 3 AI rh ~ ~.6 339 70 .41 842 9 7 1. be rou nded o ff a [Jj.1 2 .s 1.2 al aO ~1 3 .1 3.340 1 2.n iza tion.0 2 .1 2.6 180 3 3 65.62 0 18 4 78 .1 2.0 ..1 2.252 1 2.0 2.9171 82 .0 175 75.61800 TRI ANGLE 1 LENGTH.0 I L EN G T H.3000 28 74 .1 3.39 64 38 82 .0644 75 .0 2.57 10 2 5 83 .3864 .38 24 9 2 77 .7580 75 .0 ioos» edge f ir II TR IA CON BREAKDOWN FREOUENC Y ~ 4 EXPANSION 1.89 3 7 .1.0 2.5 43 03 5 696 7G.9 158 78 .0 .nJ'I.1 2.o " i.1 4.607478 72 .53 3 77 1 80 .1 3.3 2227 5 75 .in 1 .7365 .2813 81 .F AX I A L AX IAL L ENGTH-F A X IA L A XIA L L EN GT H-G AX IAL AXIAL 2.4 0 89 74 81.9 3 15 TRIACON BREAKDOWN FREOUEN CY .34 4 5 2 7 81 . 1 1.9348 .tlT .5859 79. 1 2.0 2 .1 2 .0 1. 3353 1. 178 1 .8781 .. 1 2. 2 ) dem on stra te s th e co ngr u ent t riang les an d str u ts In a 2v .4240 .594 5 6 5 82 .6243 76 .0031 77 .2 3. 1 2.36976 7 76 .9303 82.9 '158 .1 1. 1 b0i ng a rLIh"' ''' t h u"' 1 s tr e t chedn o rn e : a n less tha n 1 ~ JlJuashed }.3366 82 .0 .1 2.400996 8 1.1 2.1 1.2 2.33 3 33 3 75 . 71364 3 62 .1 1.0 1." ~ diar[:3-V dl8q!a m & c '.5488 72.1 2.0 2. 1298 .1 1.5023 0.4904 .1 3.8883 84 .1 2.61800 IIA NG L E I L EN GT H-A A X IAL AXIAL 1.1 ~1 ~O 2. " ~ ~ . so t he co o rd ina te If IJITie s.314 6 69.o{ '" l--.~ IlOMeS Ther e are 2 d if l eren t ba sic u i " nglel in th e el l i pt ic al Ico sahedro n and an v t' br eak down Will re s u lt rn a d rf h 'T .4 12220 78 .7 13 5 TRI A CON BREAKDOWN FR EO U EN CY = 2 EXPANSION = 1.9 10 5'16 77 81129 76. 11I 1l1 " ach end of the s tru. 1n co n st r uc u on tp l ~ .~ !-~ .1 2.2 86 5 73 .5592 . .9592 81. 3987 84 . F AX IA L A X IA L L EN G T H.3257 ..l A XIA L AX IA L LE NGTH·2 AXI A L A X IA L L E NGT H-3 AX IA L AXIA L 2.1 1. 1. 25 21 77 .449176 84. 1 1d th a i the 2 1) I>SIl m angles sha.1 .15 A XI A L AXIA L LE I\IGTH · 16 A XI AL A XIA L L EN GTH · 17 A X IA L AXIAL 2. l 3.4 129 10 78 .1465 76 . 1 3.1 3.sarns .1 1. 1 3.6 1800 TR IANG LE 1 L EN GT H-A AXIA L AXIA L L EN GTH . 1 3.0 ~O EGG LENGTH ·1 4 AX IA L A X I AL L EN GT H.0 1.61800 TRIANGLE 1 "L ENG T H.5908 .0 2 .0 1.618 0 3 3 72 .2 2. 1 77 .2 2.61800 TR IANG LE 1 L EN GTH -l AXI A L AXI A L LENGTH-2 AX IAL AXIA L L EN GT H-3 AX IA L AXI A L LENGT H-4 A XIA L AXIAL L EN GTH· 5 AX IA L AX IA L L EN GTH -6 A XIA L A XIA L 2.0 3.6644 2.1 2.3 128 72 . 1 0.3811 . 1 3.9348 83.2 2.3232 .8 58 50 6 72.2 0 .7882 73 . 1 3.lO AX IA L AX IAL TR IANG LE 2 1. 1 2.9138 708042 .1 1. 1 2. 10 1 th e ax i al anul!'.445083 76 .1 1.2 2.2 158 .5156 TRIANG LE 2 L EN GT H· l 1 AXI A L AXIA L L EN GT H· 12 AX IA L AXIAL LENGTH -13 AXIAL AXIA L 3.618 on " ·' 2 4 .0 1.' '.6180 3 3 77.1 2.1 2.8 74 4 82 .0 .0 1. so me spec ral ex pa nsi o n. 1 2.~ in forma ti o n '.0883 1.1561 78 . 1 2.1 2.t!nt/e.559 2 82 .1 2.2503 78.0 L EN GT H· 7 AXIA L A XIAL L EN GTH . W t! a te~r K'n.9677 79 .0 tool s..0 2.594565 82 .1 2 . D 3.4035~ 7 1.2 2.0 2.0 1.8087 62 .1 2.0 2.5908 72. 893 7 77.0000 Z A FU TRI A CON BREAKDOWN . 1 ~1 1.2670 81 .34 0 16 1 81 .8 AXIA L AXIA L LENGT H-F AX IAL AXIA L LENGTH-G AXIAL AXIAL LE NGTH-G AXIA L A XI A L 1.714 5 78 .0 1.0 1.0 1.5144 .0554 82 .1 2.69 20 3 5 61 ..0 2.1 ~1 Th e ven: ic'-)s In thf! 2 baS IC lr ii:lngle s labele d 1 :1' t h is for t ri ac on : .Z 1S .0883 71 .53 05 77 62.0 .2 2.6379 79.6149 . 1 2.7 1364 3 75. m ay be diiiHr .418726 76.9592 .31 18 0 5 81. LENGTH· 8 AXI AL AXIA L LE NGTH -9 . If yo u need the d lh ed rel anqles.2 1.0554 8 1.l l .0 2.:\X I A L L !: N G T H.8 189 77 .0 1.1 1.4523 82 .1 2.4669 75.3 32 2 79 .1 2 73 77 .563 1 80 . I.575 5 . TherR is a pape r I model o f th e 'Lv ex pansion 0.04 2 5 79 . sta rtm g With 0. I I LEN GTH.1 2.0 2.61800 T R I A N G I.5946 2.1 2.1 2 .2 anvo ns o ne A L T ER NA TE3 R E AK D OWN F F O U ENC Y ~ 2 tE E>:I' A NSI O N . t J) ~~ fig. ftp.2 2.4 E XPANS ION ' 0 . .6308 78 .0 514 4 75. rlJlJ rep eated o n both sides o f it w ill b t he same. \ L ! LENGT H: E A X I A L.03086 77 .36 74 G5 81 8237 80637 9 2704 13 80 .1 2.e fe" to th e 3"10unt o f drs to r uon Cl lo n ~: t he zenit b-ro.2969 81 . 13 A X IA L .0 2.1 4..0 . XI A L AX IAL l EN GT H.1 1.8214 . Th e sam e sy m metries hold fo r the 4v With the ad d i t io nal fact that fo r lh e t rranq lc 1' s t he ed qe o f the ico sa (d o tte d hnes) i s al so a l in e of symmetry and th e .2476 .1 strut length labels and vertex tebols for a 4v Triscon .5990 83. (3). }w t h.A AXIA L A XI AL LEN GTH -B AXIAL AX I AL L EN GT H-C AXIAL AX IA L LENGTH· D A XIA L A XIA L LENGTH -E AXIA L AXIAL TRIA NGLE 2 LE N GTH .2193 8 1.n gles con.1 2.7135 8 1.1 1.2 line o f sy m m e tr y one 0 .30 05 36 73 .0 2.5045 78 . 8 26 2 7 1.4911 14 832419 82 .4 8 546 3 78.ment bette r 0. 1 3.0865 .44 55 78 .0428 .1 4.1 .9436 80.4932 78. 36 74 6 5 80 .1 1.0 .6 8 34 1.41 64 0 5 79. 1 2.linqs. or TRIACON ELLIPTICAL This diag ram [fig.4414 .1 2.0 0.4 0 3 54 8 82 . 1 1.0 0 .1 1.0177 .1 3. I am nl U pu bli shi ng dihed ral au qles be t! .306 3 .3 72 23 0 83 .0 2. 2 2.0 8 6 5 78 .2 .45 155 5 73. 1 2.434271 81 .1 1.72 09 3 2 72 .7 14 5 78 .0 1.25 6 2 1.2 3. 1 2.6188 .0 2.38 249 2 78.6 18 0 3 3 72 0000 72 .9701 .2 2.0 ~O ~O ~o ~O .0 2 .3 811 1.0533 . 3m 1.0 1.1690 .p i n m i" llt hal the cho rd s 10 LIJ le l t 01 ' hE! lin' 0 1 sv rn rn e tr v . 3322 .1 2. 1 2. l I LENG TH -A A XIA L AXI A L L EN GT H-S AXIA L A X IA L L EI\IGT H· C AX IA L AXIA L L E N GT H -D A XI A L AX IA L TRIA I\IGLE 2 LEN GTH -E A XI A L A XIA L LEN GTH -F AXIA L AXIAL LENGTH ·G A X IA L AXIA L 2. 1. T he 'a co an oles ca ll be cal cul ated uSing t he law o f cosm ej .34 65 13 81 .0 1..0238 .0 2.57 8 60 7 65 .8087 .0 1. 1 76 . 1 1. 31 96 .~[H] FRcrOA5I Drd III . 1b r2y ~ 81.0 at eac h end o f the t w o bas ic t ri angles.6 1 36 .0 1.41 8429 69 .74 9 689 74 .1 3. 3 .1 2.1 1.3 116 .3078 67 66 62 .38 53 .2 2.6253 .1926 76 .984 7 81. t A V:7r' " '.403 548 78 .! 1.0 A LTERNATE 8 RE AKDOWN ~ ~i' ut A LTERNATE BREA KDOWN FREOUEN C Y ~ 2 EX PANSION = 0 .4 2 072 4 80.34 53 26 83.G A X IA L A X IA L L EN GT H-G A XI AL AX IA l 1.1 2.2050 .567 3 BO.1 ..4 0 2 019 63 .04 35 .0 2.0 LJ sed 2.4348 .0 3.0 .0 -.9 312 7 ~ .Jtfse Of two (o r convc ruence .2 4. g l Q .0 2.0 2.0 2.33 33 3 3 78 .O AXI AL AXI .35 7 08 1 J 74.0 0 . XI A L L EN GT H· 16 A X IA L A XIA L 1. 12 ) and fo r v e r te x lab els use up ' to 2.8345 .0 1.7 A XIA L A X IA L .34 88 79 80 .1 3.372 2 78.58 60 29 81.l AX IA L AX IA L LENGTH ·2 A XI A L AX IA L LENGTH-3 A XIA L AXIAL L EN GT H-4 AXIAL AXIA L L EN GTH -5 A XI A L A XI A L L EN GT H-6 A X IA L AX I A L L EN GT H.58 70 72 83 . 3. Jod l ot 2.1 afte r . 1 .0 2 .0 1..8686 .95 13 74 . t b~ ones t o Ul€ rlg .18 16 81.0 30 8 6 77 .2 lati v el y 1.400996 ALTERNATE ELLIPTICAL The vert ices arc labeled the same W<lY as with the spber lcal.0 2.u s ~ they are rare l y u sed (ex cept In sun d o me t ype con structi on) and wo u ld ta ke up a lot o f space. 1 .0 3.7 560 .:! OfJ t s.29 12 22 76 .2 le y ou ~.74 3487 69 .6152 .403547 80 .360 16 2 82 . 624 3 74...0 refu l Jring .0 1.2 2.1 AI .4997 83.1 1.0554 f::. 1 1. 1 4.0 3.8 29 9 1. 2.4669 .1 .2 3.4414 65.0 1. 5946 67 .1 3.4663 80 .0 2.0 2.2 73 12 1 77.0 1. 7847 ZAFU FRE O UEN CY = 3 EXPANSION = 1.5 34 9 .0 expension .0 1.30 00 28 74. ~ ~ . 1 1.'" )Z.'Z. 1 2.0 CP11. are e deast LJ sed L EN GT H· 15 AX IA L /.33 3 333 83 .6428 60 .788 1 . l' he [" lJles con rc .1 3.64 98 34 67.5911 F R EO U ENC" EXP ANS ION · 0 .9 2 65 69.8 AXIA L AXIA L EGG L E N GTH -9 AX IA L AX IA L 3. 1 2. were f laqe is 1.2 1.250 3 For the st ru t leng th label for 2v see fig .61 8 00 TR IANG LE 1 L EN GT H. 1 lon e.34651 3 82 .63 0 8 1.4 17 8 3 5 80 .1 3.t (an ex pans ion .0.56 2 7 12 8 3.1 2. ' /f\.0 2.0 2.0000 .0 1."' B ~ :J o .4 59493 65.17 A XIA L AX I AL 1.837 3 83 . f ig .' .0 1.?> 3. 1 1. 2 7.0 0.0 0.0 0. This methoc will pro bly produce smoother arcs than the triacon chord factors we have used.35929 55.2212 A &FREOUEN CY1COSAHEDRON LENGTH -A .3132066 .1 5.1 1.2 6.0 1.0 1.99999 77.1 2.2 7.8965 164.0 3.42406?-5 0.0 3.0 3.0 7.0 3.3234 7 31.1 2. 1 2.32411 73.0111 169 .3 6.1 6.1 3.1 8.0 1.1990172 .3 6.28 597 83.3 5. 36019 54.0 5.108.2939 171.0 2.11872 53. 1 1.0 2.44669 74.1 2.2 5.0 3.0 2.0 3.00277 54.1 2.1 5.0 0.0 3.22 754 11 .17371 62.385 1750 .1 6.1 7.0 7.99017 58.3203644 82.3 6.2 3.3 7.0 0.3 6.2 6.1 6.0 3.3 1.61734 60.2 6.2 1.1 2. 'tl Cut struts il get a metri c Strut A N.1 7.1 4.0 0.0 3.2 1. 110 . 1 4.1 8.1625 8-F REQUENC Y TE TR AHEDRON LENGTH L. 1 3.0 2.6335 163..26167 70 .0 5.0 3.0 3.0 3.2204 259 .0 3.1 4.0 3.1 2.4 9861 58.1 2.1 6.0 0.1 1.10660 81.6921 1.2 0.1 3.3 203644 .0 7.0 2.2 4 . 1 4.0 2.0 0.42 750 79.1 6.0 0.2 7.0 3.3 0.1 1.0 5. 1 3.0 2.1 5.1 2.0 2.1 8. For assembly diagram see vertex identification diagram.0 2.5324 171.2287 162.1 2.1 3.1 2.0 3.0 2.32411 73.1 3.32 12440 .2 1.1 1.4 1.2 6.1 4.3640 168.2 4.20597 65.0 2.0 2.1 7.0 3.1 3.0 4.1 1.0 2.S LENGTH· C LENGTH-G LENGTH-F LENGTH -H LENGTH-J LENGTH-I LENGTH-D l.2 5.0 0.67220 40 .1 3.0 3.2 5.3 8.1 5.1 6.1 3.1 3.0 2.0 0.1 2. 1 3.1 0.1 5.1 4.1 4.0 3.7 8252 80.5781 162.0 6.1 3.0 2.1 6.0 4.0 6.23 76603 .1 2.93465 78.1 0.00334 58.2 6.2 6.0 to len gth.0 0.0 0.29 179 75 .0 2.0 2.97 555 167.1 5.28745 58.1 2.0 6.41462 82 .1 4 .4355 166.0 4.07243 60 .22 134 84.2 4.0 1.1 6.1 2.75699 80.3779644 .2 5.1 2.0 0.2 3.1 0.0 0.1 1.1 1.2 1.43574 60.252604 7 84.0 1.1 2.2 0.3 5.1374 166.3 A rad ial an to th e same crro: in st ru agains t a rna I f y ou can't fo r th is type Sh If ' we be thi A· Se tri II I CU Se ler CC St A Se wi CC se C· SC C- Se' CU Co S li 112 CU Sel Cu CO Sta .2654663 .0 .0 1.U6061 164.3 6.38710 82. p.0 5.0 3. 10660 80.3 8.2 4.0 3.1 5.1 4.1 3.02343 55.1 5.81706 62.1 4.80941 79. 1403741 .12850 54.1 5.0 4.3 1.0 6.2 5.0 4.0 3. LENGTH-B LENGTH-C LENGTH-A LEN GTH -A AXIAL AXIAL AXIAL AXIAL FACE FACE FACE FAC E FACE FA CE DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL 1. 2 5.0 6.0 2.5426 169 .1 3.1 5.7834 164.71747 88.1 5.2 4.4891 167 .0 5.2 1.2 7.1 3.0 2.1 4.0 3.1 5.0 3.44669 79 .2 6.0 1.5123 98.1 1"0 . 36200 83.2 5.74402 71.8046 172.0 2.43897 81.0 2.66244 55.9111 169.0 3.0 3.0 0.0 2.1 1.53881 57.4 2 150 70.24383 40.2759:J44 80 . We have not built any domes with this method.0 4.0 6.40113 59. 1 3.1970752 .2 A 1.55 329 80.1 5.4472 136 . 2 6.1 4.2425356 .2 1. 1 2.1 4.0 2.1 5.52971 30.2 6.4595058 77.5 1/6380 .3203644 .1 4.2 6.0 3. the dome is structurally stronger. 8084 167.0 0.0 2.20222 79 .1 1.9038 161.1 6.2 7.2 0.2 4.00000 57.0 3.2 1.Figures given below are for Class I (AI crna I.1 3.0 2.1 5.40889 58.0 00 0.1 4.0 2.1 3.1 2.0 8.70583 172.ENGTH LENGTH LEN GTH LEN GTH LENGTH LENGTH LEN GTH L ENGTH L EN GTH LENGTH LENGTH LEN GTH LEN GTH LENGTH L EN GT H L EN GT H LEN GTH EN GT H LENGTH AXIAL AXIAL AXIAL AXIA L AXIA L AXIAL AXIAL A X IA L AXIAL A XIAL AXIAL AXIAL AXIAL A XIAL AXIAL AXIAL AXIAL AXIAL AXIAL AXIAL FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FA CE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDR AL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRA L DIHEDRAL DIHEDRAL 1.0 0.0 1.2 7.0 3.0 1.2995 157 .20819 57.0 0.0 5.1 3.0 8. 1 1.36284 33 .7525 172.0 1.99999 67.4 8.1 3.L ENG TH. We have not built any domes with this method .62383 174.8081 172.2 1.1 5.2503401 .0 0.2 4.1 2.97068 67 .99999 34.33 19301 .1513 168.2 6.1 3.5 9285 63.37504 75 .2 methods and breakdowns ~Ff1 CLASS I METHOD 1 .0 0.3 3 1930 1 .0 0.1 0.2 4.34037 79.2773501 .0 2.0 7.2 7.0 1.2 8.3 7.1 3.1 3. uyna Domes uses a 4 -frequency octahedron breakdown for th eir domes.1 1.04666 62.2 7.3669588 0.1 1. Important: lA's and C'. AXIA L AXIAL AXIAL AXIAL AXIAL AXIAL AXIAL FACE FACE FACE FACE FACE FACE FA CE FACE FACE FACE FACE FACE FACE FACE FACE DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHI::DRAL DIH EDRAL DIHEDRA L 0.1 2.03260 176.1 2.22134 80.1 0.28912 82.1 4.55219 63.3 7.0 1.3 6.1 0.2 7.0 00 0. 1.84385 77.0 3.71747 82 .1 4.2 8.0 3.2 7.2 6.1 6.1 2.0 2. 1 3.1 2.0 2.2 4.1 2.1 3.0 3.1 2.9038 165.2 On this rage are Chord Factors for: various other CLASS II ME THOD 1 .1 8.1 1.1 8.3 5. 3 7.1 2.0 6.0 0.2 4.1 2.0 1. 1 3.1 1.2 5. 34601 166.2 7.0 0.2 5. 1 4.1 2.0 3.0 0.3 8.1 3.1 2.0 2.1 1.1 5.0 3.4355 164.0 4.0 1.1 0.0 1.10908 79.0 1.98877 61.33 1930 1 .0 0.1 1.1 3.0 0.2 0.1 5.03267 55.1 4.0 1.2 3.0 2.0 5.1 4.4595058 .0 2.2960307 .1 5.25 280 60. Method 2.0 4.76529 53.1 3.1 0.1 1.0 2.2 4.1 4.1 4.0 2.0 2.7447 168.2 6.0 0.2 6.1 4.1 4.1 6.1 1.0 7. 1 3.2 5.1 3.07903 85.1 8.3 5.4 1.19168 77.577 3502 .1 4.29554 56.1 1.1 1.0 3.1 3.1999068 . Make models of a given frequency in both methods to see the differences.44 72 135 .4 0.0 3.2 0.'1439 171.0 0.7351 3 56.0 4.0 2.4 1.29889 6605975 61. 1 4.0 0.08534 45.0 2.0706 2 82. LENGTH -S LENGTH-D L ENGT H-C LENGT H· E I.8138 3 164.0 3.02030 45. 1 1.9439& 53.0 2.0 1.1 6. I I ITri can).27959 63.1 8.0 .1 3.1 3.2 5.1 1.1 5.1 4.7234 173.0 0.1 5.1 2.ENGTH-E AXIAL AXIAL AXIAl.8103 169.1 1.7345 168.2 3.0 3.70583 44 . 2 5.3 .1 3.2 5.2 4.0 3.97068 59.0 0.2 6. 1 3.0 3.1 3.1 2.0 2.4 8.2 5.3 6.1 3.1 5.5 2971 61.0 8.3 7.1 4.4242 174.2910 81 .1 1.9902 172. the tr iangles are more equilateral.0 2.03260 65.5529 168.2 1.3779644 .0 1.0 3.0 1.0 3. 27 357 63.0 2.1 4.2566 169.4841 169.2 383753 .3 7.1 8.0 1.2 7.0 2.44669 79.0 3.0 0.0 2.2 4.1 6.2419696 .1 3.58 361 54.0 2.1 3.0 2.0 5.0 0.1 2.29554 69.2 1.3779 644 .0 3.3669 588 77.1 1.0 0.0 1.1 6.99019 81.1 6.1 5.99 331 169.2 7. 1 4.2 5.4652 Second stet.1 6.58052 82.44669 79.1 5.2 8.1 3.0 1. 2092 173.2 4.15477 82.0 3.73513 80.2236135 .2 5.2 8. 1 1.0 3.2 4.3203644 .1 3.33 19301 .46727 83.1 1.2 4.89616 83.0 6.1513 167.0 0.0 3.577 3502 .3 7.0 1.2 5. 40862 ~FR EQUENCY ICOS AH EDRON CL ASS II METHOD 1 .3 8.3 6.0 3.0 2.0 1.0 3.2 6.1 2.2 5.0795 167.1 2.79066 59.2 6.0 4.4 7.60298 56 .0 0.1 4.3091073 .0 4.1 2.1 7.3684 177.68360 78.1 3.26167 88.2 4.0 1.0 1.0 1.1 3.2 6.1898393 .3 7.8261 171.1 3.0 0.2 7.4101 166.0 8.1 306 167. 1 7.1 3.0 0.0 3.8762 6-FREQUENCY OCTAHEDRON LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH LENGTH AXIAL AXIAL AXI AL AXIAL AXIAL AXIAL AXIAL A XIAL AXIAL AXIAL AXIAL A X IA L FACE FACE FACE FACE FACE FA CE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDR AL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL 1.94395 60.34965 40.1 1.1 4.0 1.1 3.0 5.1 6.1 1.27816 54.1 6.4 8.1 L ENGTH·A LEN GTH -G LENGTH-E LENGTH-D LENGTH-C AXIAL A X IA L AXI AL A XI AL AXIAL FACE FACE FA CE !"ACE FACE FA CE FACE DIHEDRAL DIHEDRAL DIHEDRAL DIHEDR AL 01HEDRAL 0. 2.3 1.2 1.0 2.0 2.1 2.0 4.06490 57.3 23 47 56. 65903 80.2 6. T hey are not as round as domes generated from the icosahedron.1 2.78252 76.1 3.2 5.34 034 24 .0 0.1 3.3 8. 1 3.1 4.1 3.1 4 .29 77 25 1 .2 5.0 3. 1 3.1 5.0 3.0 2.1 3.3 6.2 6.0 2.2 4.1 3.44366 69.10466 70.1 CLASS I METHOD 1 .1 2.4 7.6929 174.00490 171.1 1.0 5.3106 168.1 0.4041944 0.0 3.0 7.0 3.1 6.01768 80.10466 52.1 4.40889 45.0 2.0 0.2 4.0 7.1 2.0 1.5 176380 .1 1.0 0.0 0.0 4.1 1. 1 2.3 1.0 2.0 0.69851 59.0 3.2 5.0 2.1 7.48806 78.0 2.32457 66.2 1.0 2.1 8.10341 80.0 1.9689 178.2 7.17540 83.88121 40.0 3.1 2.1 4.8942 168. 1 1.2 3.07062 71.31269 98.1 7.14680 61.5781 162.99787 83.5607 4-FREOUENC Y OCTAHEDRON.2 5.1 7.0 4.0 3.2 8.1 1.65558 108.03357 64.67240 83.1 6.0 4. Figures below are for domes gen re lro rn the tetrahedron and octahedron .0 2.70 356 53.3590 112 81 .249 1473 .1 3.75857 78.1 5.1 2.61 734 47.82667 69. 1 3.4388710 .0 '1.0 0.24 38 120 . 1 1.0 1. as shown on p. 3C 30 30 75 75 75 B1 C 82 ~ F R E O U E NC Y I C O SA H E D R O N CLASS I METHOD 2 0.0 2.1 3.79592 89 . 1 3.0 2.1 1.2 338277 .3 8.2 8.1 5.7906 171.1306 162.ENGTH-A LENGTH-F AXIAL AXIAL AXIAL A XIAL A XIAL A XIA L FA CE FACE FACE FACE FACE FACE FACE FAC E FACE FACE DIHEDRA L DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL 1.3466883 .0 4.1 2.1 4.3382039 83.1 3.3 7.1 5.0 8.0 2.1 2.0 3.29554 65.0 0.1 2.0 5. 1598888 .0 2.3 0.26438 119 .1 4 .0 0.36916 59.96754 64.0 4.2 8.3 7243 80.2 4.9164 167.1 LENGTH-E LENGTH-C LENGTH-D LENGTH ·S LENGTH-A A XIAL AXIAL AXIAL AXIAL AXIAL AXIAL FACE FACE FACE FACE FACE FACE FACE FACE FACE FACE DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL DIHEDRAL 1.1 2.0 3.2 3.1 5.2 2.1 2.3659 168.62160 172. 6.99999 85.0 2.0347 1 81.0 0.1 3.0 3.86596 71.0 2.274 77 09 .0 2.0 1.0 3.0 1.1492 169.19075 61.31862 57.1 2.2 1.0 2.97526 84.1 5.0 3. The octahedron forms a natural t runcation at the hemisphere.1 2.0 3.1 4.1 6.4388710 .1 2. Y 0.0 0.1 Start with s tha n final Ie margin if yc On ce the ar Cut eno ugh .0 3.7B252 76.3 8.0 3.61066 80.0 1. but you do not get the smooth arcs of Method 1.0 3.0 6.1 4.0 0.1 7..3 7.0 2.26350 83.05105 80 .8103 169.3 8.0 3.36587 57.79066 45.0 3.0 3.35585 40 .----- -" ~FREQUENCYICOSAHEDRON CLASS I METHOD 2 .3 8.1 3. 39701 53.1 3.3 6.36916 52.1 1.54741 71.0 4.0 6.34503 82.0 5..1 7.2 6. Edges of the icosa face ar divided into equal parts.0 0.7990 162.07903 74 .0 1.1 1.23 119 14 . Tlbl.0 0.76529 69.2 4.0 5.0 2.4 8.78252 89.1 2.1 7.35055 60. 2 8.48972 55.4714 171 .1 4.40889 56. 1 3.02343 58.2 8.0 0.0 2881 80.0 4.0 2.1 2.0 0.0 2. 1 3.10660 84.1 7.1 86 2 170 .0 4.2 3. Figures below are fo r CI Method 1.1 2.1 3.1 5.2 1.1 2.0 3.98321 58.75086 58.2 7.0 4. 2 1.0 1.3 6.1 3. .0 0.. A ll the rest ilre the same."1J 17/ 3 2 . TO C (' .. ' c 19/ 3 2 = .75000 3/4 & -0 (]) (]) 25/ 32 = .~~7 D '10 0 I b.320 364 80 . 0637 .... :.hypotenuse c 0. 12 ? by 4 2 by ti z b. o f Frac t ious ~ . d WC 0' This form u la Can be used in che cki ng domes which are not c om p let e spheres: consider th e open bottom as a sing le face polygon.4659 .l 1 AXIAL AX IA L AXIAL LE NGTH .. 1 Where a l i ne is passed parallel to t he base of a Iriangle at o d istance Q Irom t he base the n : Ierne : 3. 1 T he t ri go nom et r ic f unctions may be lound graphi cally w ith the aid of t he foll o w i ng d iagram : .776522 76 .8 7 500 7/ 8 29/3 2 .0 2.0 1. 1 1.3987 . 1 14 IB lfJ :l!31 h 28 35 42 IB~] 16 'h 10 / 13 1 3 ta.32 69 4 6 76.1 3.0 6 2 5 0 1/1 6 = ..14 1:J926:> Circumference of circ le Area of c irc le Ar ea of sphere (sk in) Volume of sp here 2nr T 2 U 4m 2 sin 8 b ~ sine c Hal f of t his str etched octahedron can be pu I toget her With the spherical 4·frequ ency oct ahed ro n on the opposite page.15 AX I A L AX lj\ L LE NGTH -16 AXIAL AXIAL 2.4 3750 .0 3. INStDE IT TO DRAW ELUf'SE ~-.1 3. ~ T HE R END " ...QX LeN GTH F~O "'" B.0 1.----' .0 27/32 = .0 f>..6 77 0 09 80..1 4.• .0 .-0 . 0 4 ..0 2.. ECNT To-r1E\'\-'. 1 'e cot . 4 ) by 6 T he trig function s .:I:N"U ~ i:1r:J. 11/ 32 = . 2bc cos A b c 2 2 r = radius b = base h = altitude 4/3Tl r 3 ~ ~..0 4.2 5000 1/4 '" 0 E u = . All al te r n at e brua kdown is used.8 1250 (]) '" '" "'.£Ep'osit~ = )b I(-?C- _.0 2.2ab cos C ~ .1 3. / // hj~ . 0>" 5/8 = .0 2.0 3.<: .2 3. --~ -- \ft MA JO R A)l IS ~r .0 3..2 3. 4. 1 1. T here are paper mode ls of bo th 01 th ese oc t ahedrons o n p .7 33 2 14 79..2 4.90 6 25 15/1 6 '" ..1 ~K-='-BF Ec ..1 3.... 1875 . the number of laces (Fl . 53 60 12 72 .5 5 278 0 76 .. 7174 73.1 4. Gel on e at a drafting st a re..469976 74 .0 2.5005 72 . 8 2 br 10 2 by J2 3 by JO 3 bIll . 3 72 3 75 .13/1 6 = .64 2 227 77..15 6 25 5/ 32 u (]) " .40 6 25 c ~ 7/1 6 = .( ~ side ndE~..f. Lawai co sines: The square of th e length 01 a side of a tr iangle equa ls the sum of the squares of the lengths o f the o t her two sides m inu s t w ice the produ ct of th ese tw o sides times tile cosi nes of the ang le between them..34375 u a c.0 1.59375 '" .0 csc l ~ h.G o r EXAC TLY HA LF O" TAN ..8569 81 ..3557 77.-.2726 840863 OB LIOUE TR IANGLE TRIGONOMETR IC FU NCTIONS R ight-angled t r i go no m e tric [ u n c t io ns: L ~~ lI Nl: ~ E S) SIlt I t _r.>p.OCTAH E D RON EL L! PTI CAL Using the octahedron means that there is on ly one pri nciple triangle invo lved . J ++ 3. c ! f it 4..28084 ft .28 12 5 9/3 2 ~2 . 1 4.60149 1 80.0 cos " ~ side adjace~ " hy po ten use tan " Side adiacen t ~ c ~ ~~':.53125 '" co u ~ 9/16 = . F unc ti o n sin J by ~ I by 10 S 6% '2 3 4 6 8 "" '!l 7 91 /j 2 lh Jlh 2VJ 4 5 1h 8 .7454 . 1 ~ a2 + b2 18 0' ~ 90" c2 No te: See simple po cketbook Trig ts b les in bibliography.0 1.7825 80 . I n eit he r case.. 1 2. :2 REPEAT AT ---------- ---..6 79898 81 . 1 3. the Volume is in creased by a factor of 8 .0 Law of sines: Leng lh s of t he side s of a triang le are proportiona l to the sin es of th e anqles opposite t hem: B~ rr ='l.18 7 50 3/ 16 v 7/ 3 2 = .__ . b 2.1 vs to 4 . In any convex po lyhedron ..:::: E o" . 7 174 76...562 50 ~ 0'> -". 1 2..0 2...0 3.A ~BG"'AZ M ~BF Angle marker useful in measuring al l those wierd ang les.387 3 .7354 79 .LL b SiflA " ~ //j \ B BD e..2 18 75 .3 7500 3/8 .~ !.7825 514223 84 . 1 3. 2 2 a + c . 116 7 . .. DES E.. 15/ 32 = . 1 3.14 AX IAL AXIA L L ENGTH ..5 77 3 50 62 .304 8 mete rs 1 rneter v 3. This holds true also in 3-0 space and may be used for calc u lat i ng cu t-o rf d ime nsions .4 4 72 13 70 .12 AX IAL AX IAL LE NGTH. ver t ices (V) and edges (E} are re lated as fo llows: F +V ~ E +2 + ..0 Decimal Equivale nt.288 1 .4299 80.2756 75 .Q.0 3 125 1/ 3 2 = . 0 e D. Conversion of fe et to mete rs/meters to fee t : 5 5WING PE WITH lOO~C1l ~"J ~IVJX ~ ~..<: .2 ·W A F DA M 4. ....._ u 13/ 3 2 = . .50000 1/ 2 0 '.6 250 0 _ 2 1/32 '" . ON IlAAOS. - 4.__.3873 70 .2 1 by 3 11/':1 '2\h / 311 1 b. .E:.12 500 1/ 8 <0 E ~ .<: .93750 3 1/32 .0 1.71 87 5 "'. -.0 4.0 4.."..8762 .. . I t can be tr uncated perfect ly zentth -to-zeruth tu pr odu ce " shape such as the pap er mode l on p .~ 1 ft = 0. _ ..46875 '" ::> c '" = .2 539 72.. C / 2 MA. the surface area is inc reased by a facto r of 4. 3.:i. the nu mber of sides eq ua ls the nu mber 0 1 members a long th e per imeter of t he do me's f ram ewo rk .00000 '" .6 56 2 5 '" u -::l 11/ 16 = . 1 4.4 59 50 5 76.Jq~ 1 ON A SOUARE.t AX IAL AX IA L ngle f rarnu LENG TH -2 AX IAL AX IAL LE NG TH-3 AX IAL AX IAL LE NG TH -4 AX IAL AX IAL 0 .2 .D = .0 3..0 1.4 7 16 78.ra the ra tio s of sid es of r ight t r iangles.o ~ :t?rFQ?ie 1 Sl ANT RU WITH lE DESIR ED NUMBER OF DIYISIONS ACROSS BOARD .13 AXIAL AX IAL LENG TH .( . 7 174 . " ia 24 30 12 D 6% 91. .4542 11 77 . Nne af sv rn m c tr v 'o lted - s tru t leng th Ieb ets Note: usc sam e vert ex lsb cts 'I S regu/itT alte r-nato TmG RIGHT-ANG L ED TRIGONOMETR IC FUN CT ION S EX PA NDED OCTA HEDRON F R EQ U EN CY ~ 4 EXPA NS ION · 1. C 4 b)" 4 6 by 6 20 25 30 I] I !. 124 or to srt high l ik e t he agg.CUT PAPEIf. 1 1.0855 .3 1250 5/1 6 .09 3 7 5 3/ 32 = .78 125 s: c '" ...84 3 7 5 ::J a -. 30 12 16 20 24 JO 16 16 3b 16 21 11 / " 26'11 ]'2 40 48 21 1 13 4K 36 4S 54 2' 54 42 2.- lENGTH A ·( PLUS ec ""--.90 18 73.!:!~.. t he o ctuhed r nnal fo rm has th e d ist ingu ishmg ab il i ty to f use to rec t il in ear struc tures.2 3. 124.5012 70 .3400 .. ~ LE NG T H-5 AX IA L AX IAL L ENGTH-6 AX IAL AX IAL L ENGT H-7 AX IAL AX IAL LENGTH-8 AX IAL AXIAL LENGT H-9 AX IAL AX IAL LENG TH-l0 AX IAL AXIAL LENGT H._.6 87 50 ~ E a >23/32 = .2539 .".1 4.9763 . i I' -'.. DR W GE OF SOUARE..1 2.96875 1 = 1. 1 3.p I" side ouoosite a..Ht IN 10 12 rut 14 16 I by.COT ANGf N l -'1 :> 3.. 0 f/ ~ ~ ~.2 g 11% 14 .2 2. I" sodeopposite a sec Jl-. • (j'h G B 0 A R 8 113 10 10'1:1 131h HI 10 ~/J 10 ].. 1 3. (]) = .::.r: c~ . MA~K MINOR AX I~ O~ OE'SlflED Elllfl S£ A·B • TWO 8R:ADS 3 DR'V E CE A ·C APART 4 TIE A L.2ac co s B a 2 Area of t ria ngle 1/ 2 bh No te: by doubling t he d iame ter of a sp he re .eo tenu~~ ~ y a 3. 2 2 " "t. 5 1.t .6 180 0 LENGT H.~.. 1 4.!!- side ad jacent side opposi te ~ Y :DLv:r:D. 1 2.2 + c . VITAL STATISTICS (for Bubble Dome) Geometry: 3-frequency geodesic. 5/8 sphere. so they should be painted silver with rust -preventinq paint. 2A H if. (Fig 2A ) Splits along the edge of the flattened tube are re jects. Try and wangle someone into doing them ana press in a machine shop. and there must be about 3/4" beyond the holes. 1/2" is not suita ble for any domes that will be su bjec ted to heavy weather co nd it io ns. but be careful not to overstrain the vise . The tubes should be squashed as flat as you can manage with your vise. Think before cutting. and clamp it to the drill press table with C-clamps. (Fig. Bigger tubes are hard to squash! Think first. 3/4" is best for most u ses. ico sa-alternate breakdown. sti ck on small pieces of angle iron (t 1/2" wide) with pu tty to enlarge the squeezing are? of the jaws. The chord factor gives the "center-of-hole to center-of-h ole" length. It is easy to work w ith and is plated. 48 Fig. the hole you drill should be 7/16 or 15/32". eyeball the flat you have just made and in sert the other end in the vise as nearly in the same plane as you can.Cutting The tubes sh ou ld be cut according to the c ho rd factors plus 1 1/2". 2// ~ "lip s" Tube framed domes have many advantages over wood for ce rta in uses. It will probably be necessary to use a "persuader" pipe on the vise handle about 2 feet long. If you are making a smaller dome. These tips will rust. a fabric tent applied to the outside of the frame and resined . so painting isn't necessary. If the dom e is to be skinned with plastic film or fabric that touches the frame. cut a Vee groove in a 2 x 4 six inches longer than th e C strut. Splits in the middle of the flat are undesirable but usable. vertex zenith Diameter: 20' Weight {not including floor) : 600 Ibs Volume : about 2600 cu It Floor area: 314 sq It Filf. Many oth er skins co u ld be tried using the basic prin ciples outlined below. fiberglass or metal sheets. J Drilling To drill the first hole in each strut. 1 I. To fa cilitate assembly and absorb errors. Plexiglas. Transparent or tran slucent panels can be made from vinyl and inflated . In the A hole screw in a 7/16" bolt that has had its head cu t off with a hacksaw (Use a Vise-qrip or pipe wrench).e. Make the cuts with a hacksaw or a tube cutter. are arks to . This will result in about a 24 foot maximum diameter in 3-frequency . but the cheapest and eas ie st to get is " EMT" electrical conduit. To prevent this. you can use 3/4" conduit in triangles whose sides are up to 4 1/2 feet long . but a smal l amount of "lips" is acceptable. Keep the tubes in separate piles." mark lUi Frame Dome tube v ise F':~. This end will automatically have the weld positioned correctly. (Fig . angl e iron jaws 2Y. 4A 21 . Hold the tube in a vise with a stop on the bench positioned so that a cut made right against the jaw of the vise will be the correct length . (Fig . it will split. Garbage can s make handy holders. Oil the vise screw threads to m ake the turning eas ie r. Using the chord factors. There will be a hole for A. but it is useful for indoor structures and small (up to 14 feet diameter ) domes. If th e tube is positioned in the vise so that this weld co mes at the very edge of the squashed tube. Band C. You get two struts from each length for making domes up to about 24 feet diameter . 4 feet is maximum where there will be snow loads. They can be skinned in several ways: a fabric tent suspended ins ide the frame by rubber bands. For larger domes you will need bigger tubes or a higher frequency. T o squash the opposit e end. 49) lube V groove ex ect depth 10 support flat tip o Fig. 1/2" conduit will bend if climbed on . holding it perfectly horizontal and centered in the jaws vertically so that all of the tube will get sq uash ed. (BASCO makes big ch eap vise s]. Conduit comes in ten foot lengths. It wholesales for about 9li/ft . (Fig. Hammering tips flat without a vise results in poor fit and a generally crappy appearance. and when obsoleted their scrap can be largely recycled. measure exectiv the correct hole-to-hole distance (that the c ho rd factor has shown to be correct} along the 2 x 4 and drill a 3/8" hole there. Squeeze the tubes horizontally in the vise . un ison. 4A) Nail a stop block across the groove so that the drill hits dead center on the flattened tube tip and the edge of the hole nearest to the tip is about 3 /4" from the tip. Again clamp the 2 x 4 to the table so that when the tube has its first hole impaled over the cut-off bolt pin at the about %" THE FRAME Any suitably strong tubing can be used. Tube frames are well suited to flexible non -Ioad ·bearing skins and are the simplest way to make a "sky break" with no sk in at all. Use a 16 or 18 tooth hack saw blade. the tube should be positioned in the v ise with the weld at 2 o'clock . file or grind off the sharp corners of the tips at this time. If the jaws are not big enough.2A ) Make a mark on the jaws 2 1/4" in from the edge as a quid e for depth o f squeeze. so you don't have t o worry about il. by squeezing them in a vise . Th e vise-squeezinq bit is tiresome. Perhaps you can get three st ruts from a length if you combine two A and a B or something like th at.3} You can flatten these with a hammer later. tube ) nail e sure lone st op block lis to use a F ig. or even plywood .4BI For 3/4" conduit a 3/&" bolt should be used.2B) Insert the tube. To drill th e second hole in each strut. A big vise . One • staples \ v ise support bloc k with V nat! gu ide . EMT tube has a weld running the entire length of it . Fi Ie the cut-off stump to a sort of point so that the first hole already drilled in the tips of the struts can easily fit ov er it . They use mineral mater ials instead of killing trees. try to size the dome to minimize wa ste. (Fig . (See "What Size?" p. The frame can be covered with a net or m esh and foamed or ferrocemented. We find that co ndu it cutting blades for a table saw are actually slower than a hand hacksaw. Small "home workshop" vise s will break. ) Squashillg Flatten the tube tips 2 1/4" from the ends. pieces of angle iron (1 1/2" w ide) with pu tty to enlarge the squeezing area of the jaws . File the cut -off stump to a sort of point so that the first hole already drilled in the tips of the struts can easily fit over it. cut a Vee groove in a 2 x 4 six inches longer than the C strut. Hammering tips flat without a vise results in poor f it and a generally crappy appearance. by squeezing them in a vise . and clamp it to the drill press table with C-clamps.to sea F(". Squeeze . 1/2" conduit w ill bend if climbed on. It wholesales for about 9d/ft. fiberglass or metal sheets. These tips will rust. 4 feet is maximum where there will be snow loads. This will result in about a 24 foot maximum diameter in 3-frequency. If the dome is to be sk inned with pla stic film or fabric that touches the frame. Many other skins could be tried using the basic principles outlined below. For larger domes you will need bigger tubes or a higher frequency. Bigger tubes are hard to squash! Think first . You get two struts from each length for making domes up to about 24 feet diameter. vertex zenith Diameter : 20' Weight (not including floor) : 600lbs Volume: about 2600 cu ft Floor area: 314 sq ft THE FRAME Any suitably strong tubing can be used. 2fl ~ " lips" Tube framed domes have many advantages over wood for certain uses. The vise-squeezing bit is tiresome. To facilitate assembly and absorb errors. (Fig. measure exactly the correct hole-to -hole di stance (that the chord factor has shown to be correct) along the 2 x 4 and drill a 3/8" hole there. 49) tube V groove exact depth to support flat tip o Fig. or even plywood. (See "What Size?" p. stick on small . Small "home workshop" vises will break. you can use 3/4" conduit in triangles whose sides are up to 4 1/2 feet long. but it is useful for indoor structures and small (up to 14 feet diameter) domes. They can be sk inned in several ways: a fabric tent suspended inside the frame by rubber bands. Transparent or translucent panels can be made from vinyl and inflated. a fabric tent applied to the outside of the frame and resined. There will be a hole for A. 3/4" is best for most uses. It is easy to work with and is plated. 1f the jaws are not big enough. the tube should be position ed in the vise with the weld at 2 o'clock . If you are making a smaller dome. (BABCO makes big cheap vises). so you don't have to worry about it. icosa-alternate breakdown. but be careful not to over st rain the vise. angle iron jaws 2 V. but a small amount of "tips" is acceptable. but the cheapest and easiest to get is "EMT" electrical conduit. sup p or t bloc k with V nail guid~ lube nail sure one stop block . Keep the tubes in separate piles. A big vise. Oil the vise screw threads to make the turning easier . (Fig 2AI Splits alonq the edge of the flattened tube are rejects. (F ig. Make the cuts w ith a hacksaw or a tube cutter. eyeball the flat you have just made and insert the other end in th e vise as nearly in the same plane as you can .4B) For 3/4" conduit a 3/8" bolt should be used. Use a 16 or 18 tooth hacksaw blad e. file or grind off the sharp corners of the tips at this time. (Fig . Band C. (F ig . Using the chord factors. To prevent th is. (Fig. and when ob soleted their scrap can be largely recycled. 4A 21 .2A) Make a mark on the jaws 2 1/4" in from the edge as a guide for depth of squeeze. Splits in the middle of the flat are undesirable but usable. The frame can be covered with a net or mesh and foamed or ferrocemented. Th e chord factor gives the "center-of-hole to center-of-hole" length. try to size the dome to minimize waste. We find that condu it cutting blades for a table saw are actually slower than a hand hacksaw . holding it perfectly horizontal and centered in the jaws vertically so that all of the tube will get squashed . l Squashing Flatten the tube tips 2 1/4" from the ends. To drill the second hole in each strut. 48 Fig. Perhaps you can get three struts from a length if you combine two A and a B or something like that. Tube frames are well suited to flexible non-load-bearing skins and are the simplest way to make a "sky break" with no skin at all. re ks One taples CUlling The tubes should be cut according to the cho rd factors plus 1 1/2". 31 You can flatten the se with a hammer later. and there must be about 3/4" beyond the holes . the tubes horizontally in the vise. Plexiglas. it will split. so they should be painted silver with rust -preventing paint. 5/8 sphere." mark De Frame Dome tube vi se Fig.1 on. 2A Fig. In the A hole scr ew in a 7/16" bolt that has had its head cut off with a hacksaw (Use a Vise -grip or pipe wrenchl. Again clamp the 2 x 4 to the table so that when the tube has its first hole impaled over the cut-off bolt pin at the about %" VITAL STATISTICS (for Bubble Dome) Geometry: 3·frequency geodesic. Hold the tube in a vise with a stop on the bench positioned so that a cut made right against the jaw of th e vi se will be the correct length. Fig. They use mineral materials instead of killing trees. Try and wangle someone into doing them on a press in a machine shop . This end will automatically have the weld positioned correctly.2B) Insert the tube.4A) Nail a stop block across the groove so that the drill hits dead center on the flattened tube tip and the edge of the hole neare st to the tip is about 3/4" from the tip . 1/2" is not suitable for any domes that will be subjected to heavy weather conditions. Think before cutt ing . the hole you drill should be 7/16 or 15/32" . It will probably be necessary to use a "persuader" pipe on the vise handle about 2 feet long. 3 Drilling To drill the first hole in each strut. I f th e tube is positioned in the v ise so that til is weld com es at the very edge of the squashed tube. so painting isn't necessary . To squash the opposite end . Garbaqe cans make handy holders. The tubes should be squashed as flat as you can manage with your vise. EMT tube has a weld running the entire length of it . Co ~duit comes in ten foot lengths. Tip s sh o u ld b e stac ke d in an o rde r t h at makes an y given triangle as level as pos sibl e. Remember that wind lift is the largest load y o u r dome will take. We u se ringbolts (with the ring insid e) so th at w e ca n easily hong things fro m the insid e o f the d om e .frequency do me doesn't sit flat on the gro und . y o u can quickly drill oil th e sec o nd holes ac cu ra te ly in the A st ru t s. Repeat for the Band C st ru ts (mo vin g the pin ). and will make the dome symmetric. plu s e nou gh to e as ily get th e nut unto. 5) Start th e drill and lightl y touch it to th e tu b e. MUST b e co m p en sat ed for w ith extra brac ing that maintains the a ngles! Before adding any skin of any description. as for d oors.8A) A hom e sewing machine will not work well unless the dom e is sma ll o r the fabric very light. S ta n the drill and m ea sure f rom the cen ter o f the first h ole over the bolt to t h e dr ill sc ra tc h to check if it is th e co rrec t hole-to-hole distanc e. Th e sea ms c an be "fin se amed" if they're h emmed first. All ow for " sprinqba ck " by bending a bit furth er th an seems right. or it will take off and fly remarkably well. Plywood s Plywood ~ woods wit th e plywo dirt. but will be diffi eu It to waterproof and wi II not fit p erfectly . (F ig. This is a drag . Do thi s in the vise by inserting the flattened tip in the vise and bendin g th e tube t o a stop block nailed to the table. For 0 3·frequency dome. ( Fig. I f it's o. I-i'g. Hypalon or Acrilan ar e best but expensive. 8B all seams are act ually straight bu t cu rv e in use . bend the A st ru t s to 10 112 0 a nd th e B ane! C struts t o 12° . For larger domes sta rt at the bottom. b u t in any ca se .k . If you ha ve drill ed accurately the do me w ill go togeth er very quick ly u sing two step ladd ers or a sca ffo ld ..) [ stv ro to arr translucer fiberglass way to in: interior. (Fig. cheap . but rather sit s on five points. BBI sew 2 rows Plexi"tlas s Domes car and still bt metal to ri The strips and bent i' at th e hub the Plex igl expansion A second l Dum on b. several panel s can be made together rather than making all the triangles. If the clo th is wide enough. Bolts The bolts sh ou ld be long enough to go throu gh six flattened tips (remem b er that th ey p robabl y va ry in flatn ess. I f flatten ed t ips h ave split acro ss the hol e. Think it out. OK for sum m er sh ad e . they shou ld b e bent towards the split side so spl it will be inside th e dom e . Assum e they are all as thi ck as the worst one you can find) and two washer s (one on each side of the st ack). 9) I sively.r. Un to about 30 teet . Mark the table and th e 2 x 4 so you ca n ch eck if the jig is moving as yo u w ork (which is d isa ster) . it is AHSOLI ITELY NEf:ESSARY to fasten th e dome sec u rely to the ground. . Bo lt t he b otto m co u rse tightl y . These skins c an be attached to the ringbolts with innertube bands cut with tin snips. The fram will also i th e tub e edge o f tt position. The t ips can now be be n t to appro ximatel y the angle that they will hav e in the fin ished d ome fram e. Accuracy in thi s bend is not important. (Fig. Different tvp cs of skin will require different stac k in g. 7 ) Thi s will vary with the tvpe of skin that will be used . Be su re to get th em with enough thread near the ne ck to tight en th e stack of tip s p rop erly . The bolt h ead s will be in side th e dome and th e nut s outsid e. a / SKINS Suspended Skin The tube dome c a n be skinned with fabric by sew ing togeth er triangul ar pan el s that are somewhat smalle r than the ch o rds of the tubes but ar e in the same ratios. Ca reless drill ing w ill res ult in maddening m isaligned hole s during assembly! Bt>lItling Tip . a weight E w ill resul that rniqt A ligh tning ground rod is also ABSOLUTEL Y "I'/o:CESSAIt Y! Connect to any hub bolt.TUBE FRAME DOME con tinued Assembly Assembl e the frame start in g at the bottom o r tor (a nd lift it as you go). Materials su ch as mu slin and c a nvas can be us ed . the edges water. You can either block up the o t h e r ten points 0 1' lower the dome on the floor so that the five low po int s are below platform level. Thin She other end of th e 2 x 4. 6) Ac curac y in drilling t h e holes is very important unless you like lumpy domes a ssembled by beating them with a sled geha m m e r. External TUBE' apply p ressu re t oward s p in wh ile drill ing An ex ter tube [rar: The skin resined . This elim in ates most error .. or even longer -which is hand y for attaching things to the dome later . In an y ca se. Ther longest). a c that is fie the panel : su p p ly stc to the tut be roll ed . p ent. the seco nd tip will b e under the drill at pr ecis el y the co rrec t place. make each st ack of a given type (hex. A p ara c hu te will fit over a qeodesi c frame. from tru ck tu bes. no suppo rts will be need ed . as you pref er. but only s t ick a bo lt through the ne xt layer so that it will be ea sy to add to. The pieces should be sewn together u sing a double needle industrial machine and D acron thread made sp ecifi cally for tent making. etc . and make a test section befor e ch arging. For p ermanenc e. and the be st seam is the "welt sea m" unless a special tentmaking machine is available. s have to ca as it can tl fr om pane Fi g. check ing each time for accuracy before drilling th e wh ole bat ch . Pr start at t~ t riangles [ is over w~ edge will (Fig . it will be ne cessary to support ALL the poi nts if the dome is to be reall y st ro ng. irreqular h e x ) the same way. A tbree. A few tries will sho w th e proper p lace . H old the st ru t w ith pr essure again st the pin to t ak e up any slac k. (Fig. Rem ember that the remova l o f an y strut ser iou sly weakens the dome! Remov ed struts. This will m ake re-sta ck inq ea sier if that sho u ld prov e n ece ssa ry. I t is w el l to ha ve them a n inch longer than tha t .s end h o ok ed to vertex Inner tu ll e band k no tred u no ern eath th ro ug h 4" p ly wood di sc #4 gromm et in canvas 4 3 2 22 F(~. Car tubes are too weak . other than heavy sn ow. Attach with #3 grommets at vert ices and short knotted ro p es through the rubber ba nd s. 6 ---1:+ liD seam seam op e ned welt scam (fr om ou tslde d ome look ing in) numbers are s tac k iog order th . 9) If a te st panel shows that there w ill be a gap th at wi ll ad m it wind excessively.: . Make the hol es in the Plexiglas oversized le. f' (/? 9 Fig. goes over ea ch joint with caulk or DumDum on both strip surfaces. These strips must be "shingled" at the hubs.r I . but might last more or less depending on sun c o nd it io ns where you live .ifer. at pport largest poi nt d own panel ove rlap s po in t up panels. 23 . Plp'xjgla~ skin icron e le Domes can be skinned with transparent Plexiglas. and still be transparent. This method will also apply to sheet metal. Thi s will re sult in a tran slu cent watert ight dome. get a bo x of "st r ip-cau lk" at an auto supply st or e (also referred to as Dum-Dum} and roll it into little ball s and sti ck it to the tube fram e between e ach rivet. (With a 3/4" tube frame. then the "point down " tri angles of each course. (Fig.r with rsten bolt . I f the panels have been cut the right size. It is very expensive. It will "boom" if thumped in a way that might drive you crazy if wind co nd it io ns were wrong. as i t ca n take just so mu ch bending. but th e result is nice. (A #8 screw is qood] This will mean a lot of screwdriver work and drilling. overlapping the triangular shee ts at the edges and particul ar ly at the hub bolts in a shingled mann er so as t o shed water. Have a sheet metal screw every 6" . the overlap along each edge will be about 3/4" and in most ca ses this will be enough waterpr oofing. lo ok inq from outdoors ts. . (F ig. o r clips for transluc ent Filon . This will allow for the huge expansion and contraction of Plex iglas wh ich w ould otherwise break the panels. Use a caulk that is fle x ible and sun resistant and that will not attack the panel material. as you can't overlap Plexiglas like you can fiberglass or metal. (Fig. but it won't be really very bright. foi' II g. Then rivet the Plexiglas to the str ips. Blow them up with dry air or nitrogen to prevent condensation. Cut the sheet s into triangles about 3/B" larger than t he tube frame s. and rivet them on with POP rivets. Try new id ea s w ith test sect ions f irst. 10 are th ~les. The strips ar e about 1 1/2" wide 20 gauge and are POP ·r iveted to the tubes first and bent into a shallow V about 14 0 lengthwise. ~"AN" befo re assembly Fi. a caulk could be applied between the panels before riveting. You will have to carefully think out how you stack the tubes if you are using plywood. Then tape the joints with 2" wide weatherproof electrical tape (black lasts lonqest]. In any o f these domes. shingling it at the hubs and dum -dumming around th e bolts .) Domes skinned thi s way ca n be insulated by applying one inch or th icker styrofoam panel s with mastic t o the inside surface. and must have a 2" border outboard of the weld for clamping. Where the struts are vertical it doesn 't matter which edge is over which . Be su re and allow for shrinkage! The skin can be sewn to the frame here and there. because you won't have to make another skin fo r the interior. A second strip. (Fig. ~k TUBE FRAME DOME External skin continued a curat elv about iously . 12) See Bubble Dome chapter. and have the pillows seal ed by a pro in a big city. Cut the plywoods with their edges exactly according to the c ho rd factors o f the tubes. This cannot be insul ated. . and puttied. also shingled at the hubs. These sk ins will leak unless resined. This means that you st art at the bottom and apply the "point up" triangles first . These pillows are fastened to the tube fram e by impaling them (shingled) over the vertex bolt tips and th en clamping them to the frame with half-round strips made from 3/4 " PVC irriga t io n pipe split lengthways in half .00 a panel which co m p ares well with F ilo n and even plywood aft e r you pay for the pai nt . Th is will stil l let a b it of light through fiberglass panels. for metal domes. Apply a heavy coat of flexible resin or foam .l/ I. If the panels buzz again st the tube frame . T/4 " for a l /B" rivet). These panels will last about three year s. It's abo u t the only practical way to insulate suc h domes. and it requires an ex t ra st r ip of sheet metal to rivet it to. An extern al sk in can be made from trian gles the sam e si ze as the outside of the tube frame as measured in several sample places.q. Th is infor mat io n has been pr oven to work bas ically. They are c u t to have the seam we ld along the inside of each tube edge. remember. Resin can be ap plied with a weighted paint roller with a hinge in th e stick to get you over the hum p . They'll be about $3. InOated panellikin A way to get a transparent dome that is insulated to an extent is to make the panels from inflated vinyl " p illow s". It could also be rolled out into long thin s nakes a nd used as caulk between the panels. Use 20 mil tran sparent or translucent sun resistant vin yl. Paint the plywood panels and let them dry thoroughly. Use fire retard ant foam . there are many small details that you must work out for yourself as you go. lit's cheap. 11) Try to avoid large "stair-steps" from panel to panel that would make taping difficu lt. Putty arou nd bolts with DUM-DUM (see below). Paint must be free of rough dirt. Thin Sheet Stock Skin The fr ame c an be skinned w ith fiberglass sheets such as "Filon" . as there is not a great deal o f c ollected experience to guide you. TO) Plywood skins Plywood panels can be attached with "one hole clips" and 1/4" bolts. and this ends up costing less in the long run.l Drill through th e sheet tri angles while they are in position. Thi s will dampen vibrati ons. cu t the sheets as large as the outside edge of the tube s in all cases. This way. Our dome weighs 600 lbs. This turned out to be the case. snow. took some 1300 sheet metal screws and three days . And it's just the beginning . permit condensati on inside th e bubb le. at a mere 3 lbs. The floor took one day . The frame w ent up in three hour s. It is r eally nifty to li e there bobbing in our body-temperature water bed and be able to see stars. wh ich featured circul ar inflated plastic windows. and. no womb.BV JIIV & Kathleen How would you make a dome that was completely tr ansparent and st ill insul ated? Glass and other transparent sheet mater ial wo uld have to be used in a double layer to get an insulation effe ct . we did the entire thing ourselv es to make sure two persons could do it alone. We also don't like the w ay it looks. Th e thing just feels super good to be in. but we store every thing und er the bed and can st ill have a relatively usef ul changeability in the space. We tho ught that wood was not appropriate anyway .. and the flexibility of the bubbles allows for expansion and contraction. We tortured this panel with thrown bricks and sharp heavy sti cks. that is. We decided to limit our design to three frequency. The many vents keep it reasonable in hot weather. Just weather out there and you inside. We are also wor king on a production version with many improvements including pop together assembly without rivets or screws. we decided to use a tube frame. We finished it off with an insulated rug and a water bed on high enough legs so that we can store all the stuf f we aren 't using und er it. frost forming. trees. and it may well be a com plet e sphere wh ile we are at it. which it does right away because we made th e shell purposely bare of all special charact er. The idea was that when yo u came into the dome the entire thing would appear t o be empty and you coul d cal l into being any trip desired. Sadly. The waterproofness is not dependent on caulk. page 22) . We looked up Vinyl Fabricators in the Yell ow Pages. We made a radial floor. hard to accompli sh. Fire. We would never use the F ilon again in th is way.. We plan to make a dome with all the panels opening in thi s manner . We had some leaks. AnvwlIV. flat Fli< J . and store all our stuff underneath th e floor. Doing it wrecked our arm s and we had no feeling in our fingers for weeks afterward s. No roof . however. HOWever. pressure! This ruled out any possibility o f stapling or battening them to wooden frame m embers. There is a fenced off area near the " door" for taking off shoes so the dome w on't be indistinguishable from the muddy fields around it. A s we work ed with the test panel we discover ed tha t though it was a perf ect t r iangle when fl at. the bu bbles. I magine being able to open the whole thing by releasing a few ropes! A dome made this way would sacrifice the pre-compressing feature that we just discussed. Assembly was easy up to a point. but it didn 't pop . Pacific Vinyl Products in San Francisco made us a t est triangul ar inflated pan el from 20 mil sun resistant vinyl electronically sealed at the edges and equipped with an air valve. 1 wh en inflated . such as a bed. D ry nitrogen also wi ll not· . the dome proved too small for this play. We worked out a clamping strip to hold the bubbles to the tubular struts (see photo. occasionally. . That would be expen sive. As a test. th is is IT. it became invo luted into the shape seen in Fig. As we war ked on th e design it became app arent that the bubble panels trying to deform th e fram e could be a great advantage. Next time POP rivets. at least onc e the unfamiliarity of the dome itself has worn off. no hiding place. We ex posed it to h eat and cold with no serio us eff ects. The main thing is the super feeling of being alm ost outdoors. For this reason we decided to inf lat e the panels with inert nitrogen . accessible through a variety of trap doors. the character of the pl ace could be in stan tly changeable rath er than the usual home thing of having your house a sort of museum of mementos from yo ur Mexican trip or funky old immobi le and ultimately bor ing relics . Certainly you notice people m ore in our dome. because a four frequency would use much more tube than was needed for strength . For country living. and trapped moi sture would condense between the lay ers. The dome itself is only a with in-ness inside which we do ou r thing of that hour. but is absolutely solid in high buffeting winds. In the ph otographs you will see that the bottom course is skinned with Filon fiberglas riveted on . We thought that this would nicely balance the lift ~lenerated by win d . did damaqe th e vi ny l even tho ugh it w ill not suppo rt burninq. spectators. We chose the 20 fo ot size because it was the largest dome we could make in three frequency. as this load is about 2 1/2 times the wind force push load. Th is "tendency" turned out to be somewhat m ore than 200 lb s. We pumped them up from a tank of nitrogen in about an hour of deli cate hissing and one near explosion . moonl ight. But there wo uld be a huge force trying to squeeze the dome in . It's hard to imagine getting mad in one. birds. as it was more economical for this size dome. . I t booms in wind . th at was stored underneath. There would also be severe sealing probl ems and possibly even darnaqe to the skin arising from expansion and cont ract ion. As with all domes it's easy to heat. and condensati on f orms on the insid e in certai n weath er conditi ons. The translu cent Filon wo uld hide th e stuff from people outside the dome. due to the maximum size of vinyl available. About the same time we were thinking about these problems. Next we plan to try an all -openinq one. i t would tend to extingui sh any flame punctu ring the ski n. particularly in th e top pent opening. in stalled flat. the struts would be uncler a great compress ion load. but this cou ld be compensated for by using heavier tubes. rain pelting. This was done because we intended to have a second floor at the level of the top edge of th e Filon. and also a method of opening the entire top pent by means of springs. but w ould let in light. Each bu bbl e would be balancing the other bubbles alongside so there w ould be no distortion of th e st ru ts. we visi ted Philo Farnsworth and saw models of his proposed spherical dw ell ing on a pedestal. Fiv e t ries and two weeks later we finally achieved compl ete waterproofness even in violent storms. 00 I o ften th in k of a do me as a boa t haul ed as ho re and tu rn ed upsid e down. Hold them w ith one hand and reach back between you r legs with your other hand to grab the lo wer li ne..00 5.00 mi sc..00 3 rolls bu il ding p aper 9.. Mnterials ontv-sdoes not include labor. H oo k th e carabin ier through all three loops and w iggle the whole bu siness up to waist level.. you'll fall out of the harness. one -"1 ~1q'~'J'~~ L r 1& 3 ee1. 1~~'alCI€· IQI~~~ SMltJfd ~AU­ rv. . plydomes are ma de of bolted-t ogeth er sheets 011 /4 " or 3/8 " plywood . (.00 one bubble skyl ight and on e bubble w indo w 75:00 Plex iglas for w indows 14.. m endi ng str aps and m isc. you take we ight off t he k not.~\ \ ~. . a dom e has a ma st . . . Ti e end s together to f orm loop li ke this: < 0 It' s best to take loop one around again and come 90 through loop two again-for double protecti on. Cal if.00 $ 234. -... Iso ost I Pac iti« Dome 24' diameter 5/8 sphere plywood dome Floor area: 452 sq ft V olu me: abou t 4500 cub ic f t Floor fr am e pl ywood F lo or cost struts: total about 80 0 lin It @ 5380/ 1000 Bd . etc . Costs per cubic fo ot are a far better measure of value.4 The mast goes at to p center.. 6'· or rno re : _ __ --"" v 2 .8/T ie fldt fp:1 llOtA'S .. nu ts.00 26 . --. He has $ 100 invest ed in clim bing equi p ment. "" 'JI.25 Simpl e.5/8".> T h is kno t wi ll sl id e when no pre ssure (we ight) is placed on it . 2 x 6 and 4 x 6 joists and girde rs.00 20. T ot al cub ic lee t in a dome d epends upon w hat por t ion 01 a sphere you build : con dui t vi nyl pillows infla ti on (ni t rogen) spri nqs.00 nail s. PT. cheap. insula tio n To tal cost 'any $ 110. F ilon panels alum angle PVC pipe fo r cl amps lightn ing rod mise. and yo u can w ork w ith bo th hands f ree.t 1/ .y Co st s per squ are fo ot are decept ive. 00 225. I PIlJv.J04~ ~!S1~~'T \ .. .. lean back .OO Pod Dome 15' 5 1/ 2" d iameter. bol ts. It pl ywood: tota l abou t 30 sheets @ $8 nails hubs str aps. Have care not t o slip feet up-head down .... hard wa re.~ '.00 20. _~9_-. there is no separate internal framew ork. / .(.. sent us these plan s: ~ l~~ 1 I . It' s a'stra nqe sensat ion-you'll gradu ally learn to tru st th e r ig.00 96. .00 3 square s #3 red cedar shi ngles @ $ 14 42 . _..' I Carab inier w ith safety loc k : ~ Harness: You can buy a ready -made harness or make one ou t of ab out two y ards of nylon st rap from a mou ntain climbing shop.~.. Bill Woods t aught 100 LIS t h is met ho d .It: ." Z.e 5/8 Here are cost s for 1969· 70 o f t hree d if feren t types of d ome s we have bu il t . grade pl y w o od @ $7 84 . 6 sheets @ $8 48 .0 0 20.00 $664.~ '~ / r ~. 350 ...vas here ne Bubble Dome iere 20' dia meter 5/ 8 sphere vinyl p ill ow dome Floor area: 314 sq ft V ol ume : abo ut 2600 cu ft Flo or abo ut $ 160.00 240.of ing / -"\ ._-.34iE-VI'l. T o move.. -~~~~ ~IZ. proj ects 6 " or more abov e th e do me.jJ.. QQ $9 29 .~O)\ '" ---30SHEm\ \ .00 5.00 35. !~. It works best t o start at t op. I . A s with the Pod Dome. See also Fuller's patent on plydomes.75 5 . of I to . staples. hold the ent ire loop horizontally behind y ou r ass and bring the loop ends together in front of your crotch.ll11ll!1tI ~'~ . When you lean back .00 35.00 25 . 00 65. Window details . $ 14 s. ydomes .k:. Edelreid Perlon is better ..~{" _-. buck I es v inyl caulk pain t w o od for win dow batts m isc. To get into it. Try it in m odel form first . like a shi p. We made o ur m asts like th is: 5 h ol es fo r st raps t:J 00 »c : -.00 15.- ~_ -:» \~. You should pr ime and paint plywo od carefully bef ore assem bly .: .25 12 sheets 1/4" x 4' x 12' ext .00 30 . of was . and ar e no t con f ined to th e fl oor. and hold y ou as soon as you put your weight on it. It . A bout one yard of a slightly sm all er d iam eter rope (or ny l on strap ). -. but i t look s as i f leak age would be a maj or pr oblem . 00 25 .~ les f my F<>R If-11''-'''II In . 00 160. .00 20 . especially since in domes you usuall y build lo fts .1-'!-• (.NA.it' s a 9 W"V down.00 124 . and (J ives you som ethin g to throw a rop e over w hen climb ing on t he ex ter ior (wh ich will be often) ... " 0 1 Vi Pr. I I *' T he 6 " or more par t is what you th row t he rope over . say . Pull thi s line between your legs to m eet w ith the o ther loops in fro nt.00 19Id· co :ol m . \ I> .00 ~O . The ends are tied t o make a continuou s loop .?=. ~ 01 /oJr~' I CQM. F ig N ewton s T o tal cost 1 . . -. 4 x 4 redwood posts. . "'lI1t..00 _ 12.-.{. should be carefully considI ered.. but will t ighten.00 seven concre te pi ers @ . approx 35 .00 ru bber mou ld ing for di tto 18. and work you r way down : y ou get so you learn the amount of reli ef needed t o descend . ~ L oop 1 I ~ I I \ .00 30 . and soon you 're walk ing up and down on th e domes kin.k.'" 0 I t' ~I ~ . - '* -We've never built one . the k no t holds. . bent-over plywood dom e Flo or urea: 185 sq ft V olu m e: 7 1 1/8" tongue and groove plywoo d for fl oor.. People from Canyon .'1". . ~ PLYOoME I 4 ' . When y ou get t o desired po sition . bolt s. . . Ply · wood is shingled so water will run off . and either cau Ik or use a neo prene washer where bolting l t ogether. qu ick t o bu ild.. incl uding • fi I edges and bolt holes.~. . Lo ok in mountain climbin g catal ogs for the equipment: a good 1/2 " nylon rope-Goldline is o.. '- By uac inq these pee ls you can m ak e the models pictu red.. t h en t ap e th e gm her ing an gles toget her to for m curved sections. fo ur t imes fo r th e oc tahe d ron b used domes .. incl ud ing t h e gathering an gles. n "Stuff) p-'f~h vJ ~ tV! PI~ • • m~Ke 13h iccsa. the. T rac e the peel fiv e times fo r the icosa hedron based domes. ? ft ~: . ll~ ~ .. Cu t o ut. TI lI1 u m . tn. ~5Fherethis " " IS or score. ~ ?pcn d I I . T he n tape all th e sect ions toge t he r to m ake the m od el. lea ving o ne g(lth er ing angle aro und each vertex . net'''~ ?tUff) ~ lnt:o cA\r-3~a (use. f igu re t he leng ths fr om ch o rd fac to rs. " -top vertex e pe€'\ 11) ' car~.. t he n d raw the va riou s lengt h s w ith a com pass. {e iving d in cut th e ngles fr-orn ICOS1L m~ S .. To dr aw ~ t um plat e o f your ow n. ~1 iet her . . '.\ 1 ~\ ./ \ / ..I / \ c : I I . r / . i \ \. t ./ . . /. v . " . ! ".. ' '12 sphere .v~x \ .~ . .' I '\ . • t • u 'sau 5 . 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