Sphere packings, crystals, and error-correcting codesDenis Auroux January 2004 What is the densest packing? 1 Atoms in crystals Thc two dcnscst jackin¸s “tacc-ccntcrcd cu¦ic” “hcxa¸ona¦ c¦osc jackin¸” 2 Mathematical definition of density a δ ¦im a→+∞ vo¦umc ot ¦a¦¦s vo¦umc ot cu¦c 3 Examples: simj¦c cu¦ic δ π o 0.`23o ccntcrcd cu¦ic δ π √ 3 S 0.oS02 tacc-ccntcrcd cu¦ic hcxa¸ona¦ c¦osc δ π 3 √ 2 0.¨!0` 4 Various packing problems in any dimcnsion finitc jackin¸ (comj¦ctc¦y ojcn) tor lattices (trans¦ation invariancc) ? ¦atticc not a ¦atticc 5 Kepler’s conjecture (1610) `axima¦ dcnsity ot a sjhcrc jackin¸ in 3-sjacc is π 3 √ 2 0.¨!0` ( tacc-ccntcrcd cu¦ic ¦atticc) lrovcd ¦y Thomas La¦cs in 199S ln thc j¦anc ojtima¦ dcnsity is π 2 √ 3 0.90o9 (jrovcd ¦y Thuc, 1S92, 1910) ln dimcnsions !, `, o, ¨, S, dcnscst lattices arc known Lcst known jackin¸s arc ottcn ¦atticcs, ¦ut not a¦ways (c¸ dims 10, 11, 13) (Reference: N.J.A. Sloane, The sphere packing problem, http://www.math.uiuc.edu/documenta/xvol-icm/13/Sloane.MAN.ps.gz) 6 Various 3D conjectures • Icj¦cr con,ccturc (La¦cs 199S) `ax dcnsity ot a sjhcrc jackin¸ π 3 √ 2 0.¨!0` • Iissin¸ num¦cr con, (`cwton) (van dcr \acrdcn–Sch uttc 19`3. Lccch 19`o) A sjhcrc can ¦c in contact with at most 12 othcr sjhcrcs • Lodccahcdra¦ con,ccturc (Ic,cs T oth 19!2) (La¦cs-`cLau¸h¦in 199S) \o¦umc ot \oronoi cc¦¦ ≥ rc¸u¦ar dodccahcdron (⇒jackin¸ dcnsity ≤ 0.¨``) La¦cs’ jroot ot Icj¦cr uscs Ic,cs T oth rcducc to ojtimization jro¦¦cm in finitc¦y many varia¦¦cs (¦oca¦ confi¸urations – Ic,cs T oth wci¸htcd avcra¸cs ot ad,accnt \oronoi cc¦¦s) Comjutcr ca¦cu¦ations (to c¦assity ¦oca¦ confi¸urations. to so¦vc ojtimization jro¦¦cm) (Reference: T. Hales, An overview of the Kepler conjecture, http://arxiv.org/abs/math.MG/9811071) 7 Laminated lattices Stack ¦aycrs ot (n −1)-dimcnsiona¦ ¦atticcs (in “dccjcst ho¦cs”) Laminatcd ¦atticcs arc thc dcnscst oncs uj to dimcnsion S dimcnsion 1 dimcnsion 2 A A A B B 8 dimcnsion 3 A ⇒ A B ⇒ A B A A B C hcxa¸ona¦ c¦osc jackin¸ ALALALALA tacc-ccntcrcd cu¦ic ALCALCALC (on¦y ¦atticc) irrc¸u¦ar jackin¸s c¸ ALALCACLA 9 dimcnsion ! “octahcdra¦ sitcs” (natural location of doping atoms in f.c.c. crystals) 1 st ¦aycr (0,0,0,0) (0,1,1,0) (1,0,1,0) (1,1,0,0) (2,0,0,0) (2,1,1,0) (3,0,1,0) (3,1,0,0) (0,2,0,0) (0,3,1,0) (1,2,1,0) (1,3,0,0) (0,0,2,0) (0,1,3,0) (1,0,3,0) (1,1,2,0) 2 nd ¦aycr (1,0,0,1) (0,1,0,1) (0,0,1,1) (1,1,1,1) (3,0,0,1) (2,1,0,1) (2,0,1,1) (3,1,1,1) (1,2,0,1) (0,3,0,1) (0,2,1,1) (1,3,1,1) (1,0,2,1) (0,1,2,1) (0,0,3,1) (1,1,3,1) 3 rd ¦aycr 1 st ¦aycr + (0,0,0,2) 10 Coding theory (N, n, k)-codc choosc 2 n ¦inary words ot ¦cn¸th N ({0, 1} n →{0, 1} N ) so that Lammin¸ distancc ¦ctwccn va¦id codc words is ≥ k (⇒ can dctcct and corrcct | k−1 2 | crrors) (Ajj¦ications data stora¸c , communications ) lcriodization ⇒codcs can ¦c a sourcc ot dcnsc sjhcrc jackin¸s (: vicc-vcrsa) (thou¸h most common¦y uscd codcs jo¦ynomia¦ intcrjo¦ation , finitc fic¦ds) 11 Examples: dimcnsion 3, tcc ¦atticc is a (3,2,2)-codc 000,011,101,110 (3 ¦its 2 data ¦its + 1 ¦it to dctcct a sin¸¦c crror) dimcnsion !, dcnscst ¦atticc is a (!,3,2)-codc 0000,0011,0101,0110,1001,1010,1100,1111 (! ¦its 3 data ¦its + 1 ¦it to dctcct a sin¸¦c crror) dimcnsion ¨ (¨,!,3) “jcrtcct Lammin¸ codc” 0000000, 0001110, 0010101, 0011011, 0100011, 0101101, 0110110, 0111000, 1000111, 1001001, 1010010, 1011100, 1100100, 1101010, 1110001, 1111111 (¨ ¦its ! data ¦its + 3 ¦its to corrcct a sin¸¦c crror) dimcnsion 2! Lccch ¦atticc (Λ 24 ) thc dcnscst known jackin¸ in dim 2! (and in ncar¦y dimcnsions, ¦y takin¸ scctions or stackin¸ ¦aycrs) lt has thc hi¸hcst jossi¦¦c kissin¸ num¦cr (cach sjhcrc touchcs 19o`o0 othcrs') lt is a ¦aminatcd ¦atticc ¦ut can a¦so ¦c constructcd trom a (2!,12,S)-codc 12 Bibliography: • N.J.A. Sloane, The sphere packing problem, http://www.math.uiuc.edu/documenta/xvol-icm/13/Sloane.MAN.ps.gz • J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups, Springer-Verlag, 3rd edition, 1998. • T. Hales, An overview of the Kepler conjecture, http://arxiv.org/abs/math.MG/9811071 • G. Szpiro, Kepler’s conjecture: How some of the greatest minds in history helped solve one of the oldest math problems in the world, John Wiley & Sons, 2003 13 HOMEWORK Problem 1. The purpose of this problem is to verify that the density of the hexagonal close packing is equal to that of the face-centered cubic lattice (δ = π 3 √ 2 ). Consider the hexagonal close packing with spheres of radius r: a) Express in terms of r the distance h (in the “vertical” direction) between two consecutive layers of spheres. (Recall that inside each layer the spheres are in contact with each other, and that the successive layers rest on top of each other). b) Use your result to find the volume of the prism with hexagonal base and height 2h shown in the picture; deduce the density of the hexagonal close packing. Problem 2. The goal of this problem is to find the density of the laminated lattice packing in 4-dimensional space (see slide # 10). a) The 4-dimensional ball of radius a centered at the origin is the set of all points (x, y, z, u) whose distance to the origin d = x 2 + y 2 + z 2 + u 2 is less than a. Similarly to the familiar case of the 3-ball, its volume V is given by the iterated integral dV = a −a √ a 2 −x 2 − √ a 2 −x 2 √ a 2 −x 2 −y 2 − √ a 2 −x 2 −y 2 √ a 2 −x 2 −y 2 −z 2 − √ a 2 −x 2 −y 2 −z 2 dudz dy dx, Prove (by evaluating the integral) that V = 1 2 π 2 a 4 . (Hint: switch to 3D spherical coordinates after evaluating the inner-most integral! You will eventually obtain a single integral which can be evaluated by making the change of variables ρ = sin t.) b) Consider the laminated packing of 4-space: how many balls are contained in the 4-cube of side 2 whose projection is represented on slide # 10 ? Count those balls which cross the boundary with a fractional coefficient according to how much of the ball lies within the cube. (The cube is given by 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2, 0 ≤ u ≤ 2.) Remark: although balls are counted with fractional coefficients, your final answer will be an integer. Optional: explain a more efficient way of obtaining this answer. c) Show that the density of this packing is 0.617. (Find an exact formula.) (Note: the radius of the balls is not equal to 1; otherwise the balls centered at (0, 0, 0, 0) and (1, 1, 0, 0) would interpenetrate!)
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