In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid,[1] or tetrahedral pyramid. It is a 4-simplex, the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base. The regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol 3,3,3 . Contents 1 Alternative names 2 Geometry 2.1 As a configuration 2.2 Construction 2.3 Boerdijk–Coxeter helix 2.4 Projections 3 Irregular 5-cell 4 Compound 5 Related polytopes and honeycomb 6 References 7 External links Alternative names[edit] Pentachoron 4-simplex Pentatope Pentahedroid (Henry Parker Manning) Pen (Jonathan Bowers: for pentachoron)[2] Hyperpyramid, tetrahedral pyramid Geometry[edit] The 5-cell is self-dual, and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1(1/4), or approximately 75.52°. As a configuration[edit] The elements of a regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees. The self-dual simplex is identical by this rotation.[3][4] [ 5 4 6 4 2 10 3 3 3 3 10 2 4 6 4 5 ] displaystyle begin bmatrix begin matrix 5&4&6&4\2&10&3&3\3&3&10&2\4&6&4&5end matrix end bmatrix Construction[edit] The 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. (The 5-cell is essentially a 4-dimensional pyramid with a tetrahedral base.) The simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (τ,τ,τ,τ), with edge length 2√2, where τ is the golden ratio.[5] The Cartesian coordinates of the vertices of an origin-centered regular 5-cell having edge length 2 are: ( 1 10 , 1 6 , 1 3 , ± 1 ) displaystyle left( frac 1 sqrt 10 , frac 1 sqrt 6 , frac 1 sqrt 3 , pm 1right) ( 1 10 , 1 6 , − 2 3 , 0 ) displaystyle left( frac 1 sqrt 10 , frac 1 sqrt 6 , frac -2 sqrt 3 , 0right) ( 1 10 , − 3 2 , 0 , 0 ) displaystyle left( frac 1 sqrt 10 , - sqrt frac 3 2 , 0, 0right) ( − 2 2 5 , 0 , 0 , 0 ) displaystyle left(-2 sqrt frac 2 5 , 0, 0, 0right) Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2√2: ( 1 , 1 , 1 , − 1 5 ) displaystyle left(1,1,1, frac -1 sqrt 5 right) ( 1 , − 1 , − 1 , − 1 5 ) displaystyle left(1,-1,-1, frac -1 sqrt 5 right) ( − 1 , 1 , − 1 , − 1 5 ) displaystyle left(-1,1,-1, frac -1 sqrt 5 right) ( − 1 , − 1 , 1 , − 1 5 ) displaystyle left(-1,-1,1, frac -1 sqrt 5 right) ( 0 , 0 , 0 , 5 − 1 5 ) displaystyle left(0,0,0, sqrt 5 - frac 1 sqrt 5 right) The vertices of a 4-simplex (with edge √2) can be more simply constructed on a hyperplane in 5-space, as (distinct) permutations of (0,0,0,0,1) or (0,1,1,1,1); in these positions it is a facet of, respectively, the 5-orthoplex or the rectified penteract. Boerdijk–Coxeter helix[edit] A 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges represent the Petrie polygon of the 5-cell. Projections[edit] The A4 Coxeter plane projects the 5-cell into a regular pentagon and pentagram. orthographic projections Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry [5] [4] [3] Projections to 3 dimensions Stereographic projection wireframe (edge projected onto a 3-sphere) A 3D projection of a 5-cell performing a simple rotation The vertex-first projection of the 5-cell into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex. The edge-first projection of the 5-cell into 3 dimensions has a triangular dipyramidal envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope. The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection. The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here. Irregular 5-cell[edit] There are many lower symmetry forms, including these found in uniform polytope vertex figures: Symmetry [3,3,3] Order 120 [3,3,1] Order 24 [3,2,1] Order 12 [3,1,1] Order 6 [5,2]+ Order 10 Name Regular 5-cell Tetrahedral pyramid Triangular-pyramidal pyramid Pentagonal hyperdisphenoid Schläfli symbol 3,3,3 3,3 ∨ ( ) 3 ∨ Example Vertex figure 5-simplex Truncated 5-simplex Bitruncated 5-simplex Cantitruncated 5-simplex Omnitruncated 4-simplex honeycomb The tetrahedral pyramid is a special case of a 5-cell, a polyhedral pyramid, constructed as a regular tetrahedron base in a 3-space hyperplane, and an apex point above the hyperplane. The four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral pyramid vertex figures: Symmetry [3,3,1], order 24 Schlegel diagram Name Coxeter diagram × 3,3,3 × 4,3,3 × 5,3,3 t 3,3,3,3 t 4,3,3,3 t 3,4,3,3 Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. Symmetry [3,2,1], order 12 [3,1,1], order 6 [2+,4,1], order 8 [2,1,1], order 4 Schlegel diagram Name Coxeter diagram t12α5 t12γ5 t012α5 t012γ5 t123α5 t123γ5 Symmetry [2,1,1], order 2 [2+,1,1], order 2 [ ]+, order 1 Schlegel diagram Name Coxeter diagram t0123α5 t0123γ5 t0123β5 t01234α5 t01234γ5 Compound[edit] The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5-cell vertices and edges. This compound has [[3,3,3]] symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell. = ∩ . Related polytopes and honeycomb[edit] The pentachoron (5-cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group. Schläfli 3,3,3 t 3,3,3 r 3,3,3 rr 3,3,3 2t 3,3,3 tr 3,3,3 t0,3 3,3,3 t0,1,3 3,3,3 t0,1,2,3 3,3,3 Coxeter Schlegel 1k2 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxeter group E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = E ~ 8 displaystyle tilde E _ 8 = E8+ E10 = T ¯ 8 displaystyle bar T _ 8 = E8++ Coxeter diagram Symmetry (order) [3−1,2,1] [30,2,1] [31,2,1] [[32,2,1]] [33,2,1] [34,2,1] [35,2,1] [36,2,1] Order 12 120 192 103,680 2,903,040 696,729,600 ∞ Graph - - Name 1−1,2 102 112 122 132 142 152 162 2k1 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxeter group E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = E ~ 8 displaystyle tilde E _ 8 = E8+ E10 = T ¯ 8 displaystyle bar T _ 8 = E8++ Coxeter diagram Symmetry [3−1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1] Order 12 120 384 51,840 2,903,040 696,729,600 ∞ Graph - - Name 2−1,1 201 211 221 231 241 251 261 It is in the sequence of regular polychora: the tesseract 4,3,3 , 120-cell 5,3,3 , of Euclidean 4-space, and hexagonal tiling honeycomb 6,3,3 of hyperbolic space. All of these have a tetrahedral vertex figure. p,3,3 polytopes Space S3 H3 Form Finite Paracompact Noncompact Name 3,3,3 4,3,3 5,3,3 6,3,3 7,3,3 8,3,3 ... ∞,3,3 Image Cells p,3 3,3 4,3 5,3 6,3 7,3 8,3 ∞,3 It is similar to three regular polychora: the tesseract 4,3,3 , 600-cell 3,3,5 of Euclidean 4-space, and the order-6 tetrahedral honeycomb 3,3,6 of hyperbolic space. All of these have a tetrahedral cell. 3,3,p polytopes Space S3 H3 Form Finite Paracompact Noncompact Name 3,3,3 3,3,4 3,3,5 3,3,6 3,3,7 3,3,8 ... 3,3,∞ Image Vertex figure 3,3 3,4 3,5 3,6 3,7 3,8 3,∞ 3,p,3 polytopes Space S3 H3 Form Finite Compact Paracompact Noncompact 3,p,3 3,3,3 3,4,3 3,5,3 3,6,3 3,7,3 3,8,3 ... 3,∞,3 Image Cells 3,3 3,4 3,5 3,6 3,7 3,8 3,∞ Vertex figure 3,3 4,3 5,3 6,3 7,3 8,3 ∞,3 p,3,p regular honeycombs Space S3 Euclidean E3 H3 Form Finite Affine Compact Paracompact Noncompact Name 3,3,3 4,3,4 5,3,5 6,3,6 7,3,7 8,3,8 ... ∞,3,∞ Image Cells 3,3 4,3 5,3 6,3 7,3 8,3 ∞,3 Vertex figure 3,3 3,4 3,5 3,6 3,7 3,8 3,∞ References[edit] ^ Matila Ghyka, The geometry of Art and Life (1977), p.68 ^ Category 1: Regular Polychora ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 ^ Coxeter, Regular Complex Polytopes, 1991, p. 30. 4.2 The Crystalographic regular polytopes T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 H.S.M. Coxeter: Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1) Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) External links[edit] Weisstein, Eric W. "Pentatope". MathWorld. Olshevsky, George. "Pentachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007. 1. Convex uniform polychora based on the pentachoron - Model 1, George Olshevsky. Klitzing, Richard. "4D uniform polytopes (polychora) x3o3o3o - pen". Der 5-Zeller (5-cell) Marco Möller's Regular polytopes in R4 (German) Jonathan Bowers, Regular polychora Java3D Applets v t e Regular 4-polytopes Convex 5-cell 8-cell 16-cell 24-cell 120-cell 600-cell 3,3,3 pentachoron 4-simplex 4,3,3 tesseract 4-cube 3,3,4 hexadecachoron 4-orthoplex 3,4,3 icositetrachoron octaplex 5,3,3 hecatonicosachoron dodecaplex 3,3,5 hexacosichoron tetraplex Star icosahedral 120-cell small stellated 120-cell great 120-cell grand 120-cell great stellated 120-cell grand stellated 120-cell great grand 120-cell great icosahedral 120-cell grand 600-cell great grand stellated 120-cell 3,5,5/2 icosaplex 5/2,5,3 stellated dodecaplex 5,5/2,5 great dodecaplex 5,3,5/2 grand dodecaplex 5/2,3,5 great stellated dodecaplex 5/2,5,5/2 grand stellated dodecaplex 5,5/2,3 great grand dodecaplex 3,5/2,5 great icosaplex 3,3,5/2 grand tetraplex 5/2,3,3 great grand stellated dodecaplex v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polyt |