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 | B_{4}, [3,3,4], order 384 D_{4}, 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 C_{16}, 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 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.
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.
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.
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. |
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] |
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 R^{4}.
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.
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.
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:
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} {3,3^{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 |
The Möbius-Kantor polygon is a regular complex polygon _{3}{3}_{3}, , in 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 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]}
In B_{4}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]} |
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 D_{4} 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}.