Regular hexadecachoron (16cell) (4orthoplex)  

Schlegel diagram (vertices and edges)  
Type  Convex regular 4polytope 4orthoplex 4demicube 
Schläfli symbol  {3,3,4} 
Coxeter diagram  
Cells  16 {3,3} 
Faces  32 {3} 
Edges  24 
Vertices  8 
Vertex figure  Octahedron 
Petrie polygon  octagon 
Coxeter group  B_{4}, [3,3,4], order 384 D_{4}, order 192 
Dual  Tesseract 
Properties  convex, isogonal, isotoxal, isohedral, quasiregular 
Uniform index  12 
In fourdimensional geometry, a 16cell 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 also called C_{16}, hexadecachoron,^{[1]} or hexdecahedroid.^{[2]}
It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's polytope.^{[3]} Conway's name for a crosspolytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4cube), which it can be combined with to form a compound figure. The 16cell has 16 cells as the tesseract has 16 vertices.
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Transcription
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.
The 16cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix. This decomposition can be seen in a 44 duoantiprism construction of the 16cell: or , Schläfli symbol {2}⨂{2} or s{2}s{2}, symmetry 4,2^{+},4, order 64.
The 16cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16cell center.
As a configuration
This configuration matrix represents the 16cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
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. 
Orthogonal 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] 
Tessellations
One can tessellate 4dimensional Euclidean space by regular 16cells. This is called the 16cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16cell has a dihedral angle of 120°.^{[4]} Each 16cell has 16 neighbors with which it shares a tetrahedron, 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.
The dual tessellation, the 24cell 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}.
Boerdijk–Coxeter helix
A 16cell can be constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4dimensional ring. The 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16cell.
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:Symmetry constructions
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 s{2,4,3}, and Coxeter diagram: .
It can also be seen as a snub 4orthotope, represented by s{2^{1,1,1}}, and Coxeter diagram: or .
With the tesseract constructed as a 44 duoprism, the 16cell can be seen as its dual, a 44 duopyramid.
Name  Coxeter diagram  Schläfli symbol  Coxeter notation  Order  Vertex figure 

Regular 16cell  {3,3,4}  [3,3,4]  384  
Demitesseract Quasiregular 16cell 
= = 
h{4,3,3} {3,3^{1,1}} 
[3^{1,1,1}] = [1^{+},4,3,3]  192  
Alternated 44 duoprism  2s{4,2,4}  [[4,2^{+},4]]  64  
Tetrahedral antiprism  s{2,4,3}  [2^{+},4,3]  48  
Alternated square prism prism  sr{2,2,4}  [(2,2)^{+},4]  16  
Snub 4orthotope  =  s{2^{1,1,1}}  [2,2,2]^{+} = [2^{1,1,1}]^{+}  8  
4fusil  
{3,3,4}  [3,3,4]  384  
{4}+{4} or 2{4}  4,2,4 = [8,2^{+},8]  128  
{3,4}+{ }  [4,3,2]  96  
{4}+2{ }  [4,2,2]  32  
{ }+{ }+{ }+{ } or 4{ }  [2,2,2]  16 
Related complex polygons
The Möbius–Kantor polygon is a regular complex polygon _{3}{3}_{3}, , in shares the same vertices as the 16cell. It has 8 vertices, and 8 3edges.^{[5]} ^{[6]}
The regular complex polygon, _{2}{4}_{4}, , in has a real representation as a 16cell in 4dimensional space with 8 vertices, 16 2edges, only half of the edges of the 16cell. Its symmetry is _{4}[4]_{2}, order 32. ^{[7]}
In B_{4} Coxeter plane, _{2}{4}_{4} has 8 vertices and 16 2edges, shown here with 4 sets of colors. 
The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a complete bipartite graph, K_{4,4}.^{[8]} 
Related uniform polytopes and honeycombs
The regular 16cell along with the tesseract exist in a set of 15 uniform 4polytopes with the same symmetry. It is also a part of the uniform polytopes of D_{4} symmery.
This 4polytope is also related to the cubic honeycomb, order4 dodecahedral honeycomb, and order4 hexagonal tiling honeycomb which all have octahedral vertex figures.
It is in a sequence to three regular 4polytopes: the 5cell {3,3,3}, 600cell {3,3,5} of Euclidean 4space, and the order6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have tetrahedral cells.
It is first in a sequence of quasiregular polytopes and honeycombs h{4,p,q}, and a half symmetry sequence, for regular forms {p,3,4}.
See also
References
 ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
 ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
 ^ Coxeter 1973, p. 121, §7.21. see illustration Fig 7.2B.
 ^ Coxeter 1973, p. 293.
 ^ Coxeter and Shephard, 1991, p.30 and p.47
 ^ Coxeter and Shephard, 1992
 ^ Regular Complex Polytopes, p. 108
 ^ Regular Complex Polytopes, p.114
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 H.S.M. Coxeter:
 Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited 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". MathWorld.
 Der 16Zeller (16cell) Marco Möller's Regular polytopes in R^{4} (German)
 Description and diagrams of 16cell projections
 Klitzing, Richard. "4D uniform polytopes (polychora) x3o3o4o  hex".