16-cell

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, regular
Uniform index	12

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. 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.

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. 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

The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).

Each of its 4 successor convex regular 4-polytopes can be constructed as the convex hull of a polytope compound of multiple 16-cells: the 16-vertex tesseract as a compound of two 16-cells, the 24-vertex 24-cell as a compound of three 16-cells, the 120-vertex 600-cell as a compound of fifteen 16-cells, and the 600-vertex 120-cell as a compound of seventy-five 16-cells.

Regular convex 4-polytopes
Symmetry group	A4	B4		F4	H4
Name	5-cell Hyper- tetrahedron	16-cell Hyper- octahedron	8-cell Hyper- cube	24-cell	600-cell Hyper- icosahedron	120-cell Hyper- dodecahedron
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter diagram
Graph
Vertices	5	8	16	24	120	600
Edges	10	24	32	96	720	1200
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 tetrahedra	16 tetrahedra	8 cubes	24 octahedra	600 tetrahedra	120 dodecahedra
Long radius	1	1	1	1	1	1
Edge length	√5/√2 ≈ 1.581	√2 ≈ 1.414	1	1	1/ϕ ≈ 0.618	1/√2ϕ2 ≈ 0.270
Short radius	1/4	1/2	1/2	√2/2 ≈ 0.707	1 - (√2/2√3φ)2 ≈ 0.936	1 - (1/2√3φ)2 ≈ 0.968
Area	10•√8/3 ≈ 9.428	32•√3/4 ≈ 13.856	24	96•√3/4 ≈ 41.569	1200•√3/8φ2 ≈ 99.238	720•25+10√5/8φ4 ≈ 621.9
Volume	5•5√5/24 ≈ 2.329	16•1/3 ≈ 5.333	8	24•√2/3 ≈ 11.314	600•1/3√8φ3 ≈ 16.693	120•2 + φ/2√8φ3 ≈ 18.118
4-Content	√5/24•(√5/2)4 ≈ 0.146	2/3 ≈ 0.667	1	2	Short∙Vol/4 ≈ 3.907	Short∙Vol/4 ≈ 4.385

Disjoint squares
xy plane ( 0, 1, 0, 0) ( 0, 0,-1, 0) ( 0, 0, 1, 0) ( 0,-1, 0, 0)
wz plane ( 1, 0, 0, 0) ( 0, 0, 0,-1) ( 0, 0, 0, 1) (-1, 0, 0, 0)

The 16-cell is the 4-dimensional cross polytope, which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system.

The eight vertices 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 edge length is √2.

The vertex coordinates form 6 orthogonal central squares lying in the 6 coordinate planes. Squares in opposite planes that do not share an axis (e.g. in the xy and wz planes) are completely disjoint (they do not intersect at any vertices).

The 16-cell constitutes an orthonormal basis for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes.

Structure

The Schläfli symbol of the 16-cell is {3,3,4}, indicating that its cells are regular tetrahedra {3,3} and its vertex figure is a regular octahedron {3,4}. 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 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 orthogonal central squares lying on great circles in the 6 coordinate planes (3 pairs of completely orthogonal great squares). At each vertex, 3 great squares cross perpendicularly. The 6 edges meet at the vertex the way 6 edges meet at the apex of a canonical octahedral pyramid.

Rotations

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

A 3D projection of a 16-cell performing a double rotation

Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each of the 16-cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares). Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes (e.g. the xy plane) and another angle of rotation in the completely orthogonal great square plane (the wz plane). Completely orthogonal great squares have disjoint vertices: 4 of the 16-cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.

In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4-space is called a simple rotation, in which only one of the two completely orthogonal planes rotates (the angle of rotation in the other plane is 0). In the 16-cell, a simple rotation in one of the 6 orthogonal planes moves only 4 of the 8 vertices; the other 4 remain fixed. (In the simple rotation animation above, all 8 vertices move because the plane of rotation is not one of the 6 orthogonal basis planes.)

In a double rotation both sets of 4 vertices move, but independently: the angles of rotation may be different in the 2 completely orthogonal planes. If the two angles happen to be the same, a maximally symmetric isoclinic rotation takes place. In the 16-cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.

Constructions

Octahedral dipyramid

Octahedron $\beta _{3}$	16-cell $\beta _{4}$
Orthogonal projections to skew hexagon hyperplane

The simplest construction of the 16-cell is on the 3-dimensional cross polytope, the octahedron. The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its Petrie polygon is the hexagon). Add another pair of vertices, on a fourth axis perpendicular to all 3 of the other axes. Connect each new vertex to all 6 of the original vertices, adding 12 new edges. This raises two octahedral pyramids on a shared octahedron base that lies in the 16-cell's central hyperplane.

Stereographic projection of the 16-cell's 6 orthogonal central squares onto their great circles. Each circle is divided into 4 arc-edges at the intersections where 3 circles cross perpendicularly. Notice that each circle has one Clifford parallel circle that it does not intersect. Those two circles pass through each other like adjacent links in a chain.

The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections). Each square intersects with each of the other squares at two opposite vertices, with two of the squares crossing at each vertex. Then two more points are added in the fourth dimension (above and below the 3-dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and three more squares (which appear edge-on as the 3 diameters of the hexagon in the projection).

Something unprecedented has also been created. Notice that each square no longer intersects with all of the other squares: it does intersect with four of them (with three of the squares crossing at each vertex now), but each square has one other square with which it shares no vertices: it is not directly connected to that square at all. These two separate perpendicular squares (there are three pairs of them) are like the opposite edges of a tetrahedron: perpendicular, but non-intersecting. They lie opposite each other (parallel in some sense), and they don't touch, but they also pass through each other like two perpendicular links in a chain (but unlike links in a chain they have a common center). They are an example of Clifford parallel polygons, and the 16-cell is the simplest regular polytope in which they occur. Clifford parallelism emerges here and occurs in all the subsequent 4-dimensional convex regular polytopes, where it can be seen as the defining relationship among disjoint regular 4-polytopes and their co-centric parts. It can occur between congruent (similar) polytopes of 2 or more dimensions. For example, as noted above all the subsequent convex regular 4-polytopes are compounds of multiple 16-cells; those 16-cells are Clifford parallel polytopes.

Wythoff constructions

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.

Helical construction

Net and orthogonal projection

A 16-cell can be constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each bent in the fourth dimension into a ring. The two circular helixes spiral around each other, nest into each other and pass through each other forming a Hopf link. 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.

Thus the 16-cell can be decomposed into two similar cell-disjoint circular chains of eight tetrahedrons each, four edges long. 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.

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.

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

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°. 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.

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]

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

A 3-dimensional projection of the 16-cell and 4 intersecting spheres (a Venn diagram of 4 sets) are topologically equivalent.

The 16 cells ordered by number of intersecting spheres (from 0 to 4)     (see all cells and k-faces)

4 sphere Venn diagram and 16-cell projection in the same orientation

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	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

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.

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.

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.

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}.

See also

• 24-cell
• 4-polytope
• D4 polytope

References

• 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.
  • Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.
  • 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]
  • Coxeter, H.S.M.; Shephard, G.C. (1992). "Portraits of a family of complex polytopes". Leonardo. 25 (3/4): 239–244. doi:10.2307/1575843. JSTOR 1575843. S2CID 124245340.
• 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)
• Kim, Heuna; Rote, Günter (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].

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".