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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.[1] It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the tesseract (4-cube). Conway's name for a cross-polytope is orthoplex, for orthant complex. The 16-cell has 16 cells as the tesseract has 16 vertices.

Contents

1 Geometry

1.1 As a configuration

2 Images

2.1 Orthogonal projections

3 Tessellations 4 Boerdijk–Coxeter helix 5 Projections 6 4 sphere Venn Diagram 7 Symmetry constructions 8 Related complex polygons 9 Related uniform polytopes and honeycombs 10 See also 11 References 12 External links

Geometry[edit] 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 16-cell 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 16-cell 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 16-cell 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 4-4 duoantiprism construction of the 16-cell: or , Schläfli symbol 2 ⨂ 2 or s 2 s 2 , symmetry [[4,2+,4]], order 64. The 16-cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16-cell center. 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.[2][3]

[

8

6

12

8

2

24

4

4

3

3

32

2

4

6

4

16

]

displaystyle begin bmatrix begin matrix 8&6&12&8\2&24&4&4\3&3&32&2\4&6&4&16end matrix end bmatrix

Images[edit]

Stereographic projection

A 3D projection of a 16-cell performing a simple rotation.

The 16-cell 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[edit]

orthographic projections

Coxeter plane B4 B3 / D4 / A2 B2 / D3

Graph

Dihedral symmetry [8] [6] [4]

Coxeter plane F4 A3

Graph

Dihedral symmetry [12/3] [4]

Tessellations[edit] One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol 3,3,4,3 . Hence, the 16-cell has a dihedral angle of 120°.[4] The dual tessellation, 24-cell honeycomb, 3,4,3,3 , is made of by regular 24-cells. Together with the tesseractic honeycomb 4,3,3,4 , these are the only three regular tessellations of R4. Each 16-cell 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. Twenty-four 16-cells meet at any given vertex in this tessellation. Boerdijk–Coxeter helix[edit] A 16-cell can be constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4-dimensional 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 16-cell.

Projections[edit]

Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)

The cell-first parallel projection of the 16-cell into 3-space 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 (non-regular) 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 16-cell, all its edges lie on the faces of the cubical envelope. The cell-first perspective projection of the 16-cell into 3-space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection. The vertex-first parallel projection of the 16-cell into 3-space 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 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron. Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a hexagonal bipyramidal envelope. 4 sphere Venn Diagram[edit] The usual projection of the 16-cell and 4 intersecting spheres (a Venn diagram of 4 sets) form topologically the same object in 3D-space:

Symmetry constructions[edit] There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, 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 4-orthotope, represented by s 21,1,1 , and Coxeter diagram: or . With the tesseract constructed as a 4-4 duoprism, the 16-cell can be seen as its dual, a 4-4 duopyramid.

Name Coxeter diagram Schläfli symbol Coxeter notation Order Vertex figure

Regular 16-cell

3,3,4 [3,3,4] 384

Demitesseract Quasiregular 16-cell = = h 4,3,3 3,31,1 [31,1,1] = [1+,4,3,3] 192

Alternated 4-4 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 4-orthotope = s 21,1,1 [2,2,2]+ = [21,1,1]+ 8

4-fusil

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[edit] The Möbius–Kantor polygon is a regular complex polygon 3 3 3, , in

C

2

displaystyle mathbb C ^ 2

shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.[5] [6] The regular complex polygon, 2 4 4, , in

C

2

displaystyle mathbb C ^ 2

has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is 4[4]2, order 32. [7]

Orthographic projections of 2 4 4 polygon

In B4 Coxeter plane, 2 4 4 has 8 vertices and 16 2-edges, 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, K4,4.[8]

Related uniform polytopes and honeycombs[edit] The regular 16-cell along with the tesseract exist in a set of 15 uniform 4-polytopes with the same symmetry. It is also a part of the uniform polytopes of D4 symmery. This 4-polytope is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb which all have octahedral vertex figures. It is in a sequence to three regular 4-polytopes: the 5-cell 3,3,3 , 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 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[edit]

24-cell 4-polytope

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 polytopes and compounds

References[edit]

^ Matila Ghyka, The Geometry of Art and Life (1977), p.68 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 ^ Coxeter, Regular polygons, 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 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. "16-Cell". MathWorld.  Der 16-Zeller (16-cell) Marco Möller's Regular polytopes in R4 (German) Description and diagrams of 16-cell projections Klitzing, Richard. "4D uniform polytopes (polychora) x3o

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