16-cell

Regular hexadecachoron
(16-cell)
(4-orthoplex)
Regular hexadecachoron; (16-cell); (4-orthoplex)
	Schlegel diagram; (vertices and edges)
Type	Convex regular 4-polytope; 4-orthoplex; 4-demicube
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	B4, [3,3,4], order 384; D4, order 192
Dual	Tesseract
Properties	convex, isogonal, isotoxal, isohedral, quasiregular
Uniform index	12

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,^[1] or hexdecahedroid.^[2]

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's $\beta _{4}$ polytope.^[3] Conway's name for a cross-polytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices.

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 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

This configuration matrix represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

${\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix}}\end{bmatrix}}$

Images

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

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

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] 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.

The dual tessellation, the 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.

Boerdijk–Coxeter helix

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

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

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

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{2^1,1,1}, 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
Regular 16-cell		{3,3,4}	[3,3,4]	384
Demitesseract Quasiregular 16-cell	= =	h{4,3,3} {3,3^1,1}	[3^1,1,1] = [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{2^1,1,1}	[2,2,2]⁺ = [2^1,1,1]⁺	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

The Möbius–Kantor polygon is a regular complex polygon ₃{3}₃, , in $\mathbb {C} ^{2}$ 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 $\mathbb {C} ^{2}$ 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, K_4,4.^[8]

Related uniform polytopes and honeycombs

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 symmetry.

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}.

References

^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 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 Semi-Regular 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, 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: 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. "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) x3o3o4o - hex".

[1] N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249

[2] Matila Ghyka, The Geometry of Art and Life (1977), p.68

[FOOTNOTECoxeter1973121§7.21._see_illustration_Fig_7.2<small>B</small>-3] Coxeter 1973, p. 121, §7.21. see illustration Fig 7.2B.

[FOOTNOTECoxeter1973293-4] Coxeter 1973, p. 293.

[5] Coxeter and Shephard, 1991, p.30 and p.47

[6] Coxeter and Shephard, 1992

[7] Regular Complex Polytopes, p. 108

[8] Regular Complex Polytopes, p.114

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v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds