 16cell

Regular hexadecachoron
(16cell)
(4orthoplex)
Schlegel diagram
(vertices and edges)Type Convex regular 4polytope Schläfli symbol {3,3,4}
{3,3^{1,1}}
h{4,3,3}
s{2,2,2}CoxeterDynkin diagram
Cells 16 {3,3} Faces 32 {3} Edges 24 Vertices 8 Vertex figure
OctahedronPetrie polygon octagon Coxeter group C_{4}, [3,3,4]
D_{4}, [3^{1,1,1}]
[2^{3}] (half)Symmetry group [3,3,4], order 384
[3^{1,1,1}], order 192
[3,4,2^{+}], order 48
[2^{3}]^{+}, order 8Dual Tesseract Properties convex, isogonal, isotoxal, isohedral Uniform index 12 In four dimensional geometry, a 16cell or hexadecachoron is a regular convex 4polytope. It is one of the six regular convex 4polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid19th century.
It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes. The dual polytope is the tesseract (4cube). Conway's name for a crosspolytope is orthoplex, for orthant complex.
Contents
Geometry
It is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces, 24 edges, and 8 vertices. The 24 edges bound 6 squares lying in the 6 coordinate planes.
The eight vertices of the 16cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs.
The Schläfli symbol of the 16cell is {3,3,4}. Its vertex figure is a regular octahedron. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge.
There is a lower symmetry form of the 16cell, called a demitesseract or 4demicube, a member of the demihypercube family, and represented by h{4,3,3}, and Coxeter diagrams or . It can be drawn bicolored with alternating tetrahedral cells.
It can also be seen in lower symmetry form as a tetrahedral antiprism, constructed by 2 parallel tetrahedra in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by h_{0,1}{2,4,3}, and Coxeter diagram: .
It can also be seen as a snub 4orthotope, represented by s{2,2,2}, and Coxeter diagram: .
Images
Stereographic projection
A 3D projection of a 16cell performing a simple rotation.
The 16cell has two Wythoff constructions, a regular form and alternated form, shown here as nets, the second being represented by alternately two colors of tetrahedral cells.orthographic projections Coxeter plane B_{4} B_{3} / D_{4} / A_{2} B_{2} / D_{3} Graph Dihedral symmetry [8] [6] [4] Coxeter plane F_{4} A_{3} Graph Dihedral symmetry [12/3] [4] Orthogonal projection graphs
demitesseract in order4 Petrie polygon symmetry as an alternated tesseract
TesseractTessellations
One can tessellate 4dimensional Euclidean space by regular 16cells. This is called the hexadecachoric honeycomb and has Schläfli symbol {3,3,4,3}. The dual tessellation, icositetrachoric honeycomb, {3,4,3,3}, is made of by regular 24cells. Together with the tesseractic honeycomb {4,3,3,4}, these are the only three regular tessellations of R^{4}. Each 16cell has 16 neighbors with which it shares an octahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twentyfour 16cells meet at any given vertex in this tessellation.
Projections
The cellfirst parallel projection of the 16cell into 3space has a cubical envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (nonregular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16cell, all its edges lie on the faces of the cubical envelope.
The cellfirst perspective projection of the 16cell into 3space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cellfirst parallel projection.
The vertexfirst parallel projection of the 16cell into 3space has an octahedral envelope. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16cell. The closest vertex of the 16cell to the viewer projects onto the center of the octahedron.
Finally the edgefirst parallel projection has a shortened octahedral envelope, and the facefirst parallel projection has a hexagonal bipyramidal envelope.
4 sphere Venn Diagram
The usual projection of the 16cell and 4 intersecting spheres (a Venn diagram of 4 sets) form topologically the same object in 3Dspace:
Related uniform polytopes
Name tesseract rectified
tesseracttruncated
tesseractcantellated
tesseractruncinated
tesseractbitruncated
tesseractcantitruncated
tesseractruncitruncated
tesseractomnitruncated
tesseractCoxeterDynkin
diagramSchläfli
symbol{4,3,3} t_{1}{4,3,3} t_{0,1}{4,3,3} t_{0,2}{4,3,3} t_{0,3}{4,3,3} t_{1,2}{4,3,3} t_{0,1,2}{4,3,3} t_{0,1,3}{4,3,3} t_{0,1,2,3}{4,3,3} Schlegel
diagramB_{4} Coxeter plane graph Name 16cell rectified
16celltruncated
16cellcantellated
16cellruncinated
16cellbitruncated
16cellcantitruncated
16cellruncitruncated
16cellomnitruncated
16cellCoxeterDynkin
diagramSchläfli
symbol{3,3,4} t_{1}{3,3,4} t_{0,1}{3,3,4} t_{0,2}{3,3,4} t_{0,3}{3,3,4} t_{1,2}{3,3,4} t_{0,1,2}{3,3,4} t_{0,1,3}{3,3,4} t_{0,1,2,3}{3,3,4} Schlegel
diagramB_{4} Coxeter plane graph See also
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes References
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, p.296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (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 ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
 Weisstein, Eric W., "16Cell" from MathWorld.
 Olshevsky, George, Hexadecachoron at Glossary for Hyperspace.
 Der 16Zeller (16cell) Marco Möller's Regular polytopes in R^{4} (German)
 Description and diagrams of 16cell projections
 Richard Klitzing, 4D uniform polytopes (polychora), x3o3o4o  hex
Categories: Fourdimensional geometry
 Polychora
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