Tetrahedron

SEARCH MATHWORLDAlgebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Tetrahedron in the Geometry > Solid Geometry > Polyhedra > Tetrahedra > Geometry > Solid Geometry > Polyhedra > Platonic Solids > Recreational Mathematics > Mathematical Art > String Figures > More... Octahedron Fractal Tetrahedron Decomposition of a Tetrahedron into Four Congruent Parts Four Dividable Rhombohedra Alphabetical Index Interactive Entries Random Entry New in MathWorld Ring Structures Made of Square Pyramids and Matching Tetrahedra MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book 12,906 entries Last updated: Fri Jan 23 2009 The regular tetrahedron, often simply called "the" tetrahedron, is the Platonic solid with four polyhedron vertices, six polyhedron . It is . It is . Get your own MathWorld T-shirt » Other Wolfram Web Resources » edges, and four equivalent equilateral triangular faces, also uniform polyhedron and Wenninger model described by the Schläfli symbol and the Wythoff symbol is The tetrahedron has 7 axes of symmetry: and (axes connecting vertices with the centers of the opposite faces) (the axes connecting the midpoints of opposite sides). There are no other convex polyhedra other than the tetrahedron having four faces. The tetrahedron has two distinct nets (Buekenhout and Parker 1998). Questions of polyhedron coloring of the tetrahedron can be addressed using the Pólya enumeration theorem. The surface area of the tetrahedron is simply four times the area of a single face, so (1) Since a tetrahedron is a pyramid with a triangular base, , giving (2) The dihedral angle is (3) The midradius of the tetrahedron is (4) (5) Plugging in for the polyhedron vertices gives (6) and it cannot be stellated. 0). 201). and let its base lie in the plane with one vertex lying along the . The polyhedron vertices of this tetrahedron are then located at ( . where on a side. ( . it is sliced into 24 pieces (Gardner 1984. 1991). . and the number can be reduced to six for squashed irregular tetrahedra (Wells 1975. Langman 1951). 0. positive -axis. pp. p. pp. 0. and the complete graph . 56-57). and so can form a universal coupling if hinged appropriately. Let a tetrahedron be length ). and (0. The figure above shows an origami tetrahedron constructed from a single sheet of paper (Kasahara and Takahama 1987. The opposite edges of a regular tetrahedron are perpendicular. If a regular tetrahedron is cut by six planes. and therefore the centers of the faces of a tetrahedron form another tetrahedron (Steinhaus 1999. The tetrahedron is the only simple polyhedron with no polyhedron diagonals. The tetrahedron is its own dual polyhedron.The dual polyhedron of an tetrahedron with unit edge lengths is another oppositely oriented tetrahedron with unit edge lengths. The connectivity of the vertices is given by the tetrahedral graph. It is the prototype of the tetrahedral group equivalent to the circulant graph . 190 and 192. including kites (Bell 1903. 184-185). Eight regular tetrahedra can be placed in a ring which rotates freely. 0). each passing through an edge and bisecting the opposite edge. pp. Gardner 1984. Alexander Graham Bell was a proponent of use of the tetrahedron in framework structures. Lesage 1956. 0. 1). 192-194). 0. geometric centroid of the vertices. One such tetrahedron for a cube of side length 1 gives having vertices (0.(7) is then (8) This gives the area of the base as (9) The height is (10) The circumradius is found from (11) (12) Solving gives (13) (14) (15) The inradius is (16) (17) which is also (18) (19) The angle between the bottom plane and center is then given by (20) (21) (22) (23) Given a tetrahedron of edge length . 1). pp. 0). . the four polyhedron vertices are located at The vertices of a tetrahedron of side length the tetrahedron of side length can also be given by a particularly simple form when the vertices are taken as corners of a cube (Gardner 1984. as shown above (24) (25) (26) (27) situated with vertical apex and with the origin of coordinate system at the . 1. 0). and satisfies the inequalities (28) (29) (30) . (0. (1. (1. with. 1. and . defined as a convex polyhedron consisting of four (not necessarily identical) triangular faces. the (33) If the faces are congruent and the sides have lengths .. Then the volume is given by . . . By slicing a tetrahedron as shown above. Now consider a general (not necessarily regular) tetrahedron. 246). pp. In general. Let the areas of these triangles be and for by . . Then the four face areas are (36) . 205). result triakis tetrahedron 2 3 cube 9-faced star deltahedron Connecting opposite pairs of edges with equally spaced lines gives a configuration like that shown above which divides the tetrahedron into eight regions: four open and four closed (Steinhaus 1999.(31) The following table gives polyhedra which can be constructed by cumulation of a tetrahedron by pyramids of given heights . 191-192). Let the tetrahedron be specified by its polyhedron vertices at where . and .. . p. then the volume (35) Consider an arbitrary tetrahedron . a square can be obtained. . The projection of a tetrahedron can be an equilateral triangle or a square (32) Specifying the tetrahedron by the three polyhedron edge vectors . then (34) (Klee and Wagon 1991. 4. p. This cut divides the tetrahedron into two congruent solids rotated by (Steinhaus 1999. and connected by denote the dihedral angle with respect to with triangles . and . respectively. . if the edge between vertices and is given by the Cayley-Menger determinant is of length . and volume is from a given polyhedron vertex.. . and . Then the volume of a tetrahedron is given by (39) (P. Jackson. and . . 20. 48-110 and 250. Furthermore. Given a right-angled tetrahedron with one apex where all the edges meet orthogonally and where the face opposite this apex is denoted . Write for the complement in . Consequently.e. G. Boca Raton.. pp. . comm. Sierpiński Tetrahedron. 252-293. . Lee 1997). not necessarily regular) tetrahedron (Altshiller-Court 1979). Isosceles Tetrahedron. Geographic 44. and let be the angle subtended by edge . (the (42) or approximately 0. des candidats aux grandes écoles et à l'agrégation. Guy 1994). This is a generalization of the law of cosines to the tetrahedron. . It follows that any plane passing through a bimedian of a tetrahedron bisects the volume of the tetrahedron (Altshiller-Court 1979. Sphere Tetrahedron Picking. Balliccioni. so that be the set of the power set of and the product of . p. . comm. Then (41) as shown by J. CRC Standard Mathematical Tables. F. Couderc. 228. pp. and . Tetrahedron 6-Compound. In of an element such that span a face of the tetrahedron. FL: CRC Press. 89). Truncated Tetrahedron Portions of this entry contributed by Frank Jackson REFERENCES: Altshiller-Court. and denote the side lengths . 186. Tetrahedron 4-Compound. Reuleaux Tetrahedron. Kaeser. which associates to an element to the sum of for all . and Parker. Now define . Let let be the set of edges of a tetrahedron and . W. pers. Bell. and Balliccioni. Premier livre du tétraèdre à l'usage des éléves de première. "The Number of Nets of the Regular Convex Polytopes in Dimension . If a plane divides two opposite edges of a tetrahedron in a given ratio. (37) where is the length of the common edge of and (Lee 1997). 69-94. A. The above formula provides the means to calculate the solid angle subtended by the vertex of a regular tetrahedron by substituting dihedral angle). p. New York: Chelsea. the volume and circumradius of the tetrahedron are related by the beautiful formula (40) (Crelle 1821. Stella Octangula. Let be the set of triples . 219-251. 28th ed. A. p. "The Tetrahedron. there are therefore three elements of length the quantity for all . which associates which are the pairs of opposite edges. SEE ALSO: Augmented Truncated Tetrahedron. Buekenhout. . Beyer." Ch.-P. N. Cube Tetrahedron Picking. 1935. Paris: Gauthier-Villars. de mathématiques.55129 steradians. "The Tetrahedral Principle in Kite Structure. Tangent Spheres. Rouché and Comberousse 1922. There are a number of interesting and unexpected theorems on the properties of general (i. p. pp. The analog of Gauss's circle problem can be asked for tetrahedra: how many lattice points lie within a tetrahedron centered at the origin with a given inradius (Lehmer 1940. Tetrahedron 10Compound. 90). then (38) This is a generalisation of Pythagoras's theorem which also applies to higher dimensional simplices (F. P. Xu and Yau 1992. Ehrhart Polynomial. then it divides the volume of the tetrahedron in the same ratio (Altshiller-Court 1979.involving the six dihedral angles (Dostor 1905. M. Coordonnées barycentriques et géométrie. Granville 1991. Heronian Tetrahedron. Gua de Malves around 1740 or 1783 (Hopf 1940). 2006)." Nat. Pentatope. Polyhedron Coloring. von Staudt 1860. . Tetrahedron 5-Compound. Then if denotes the area of the triangle with sides of lengths . Hilbert's 3rd Problem. 568-576 and 643-664. 1964. 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