16-cell

Regular hexadecachoron
(16-cell)
(4-orthoplex)

Schlegel diagram
(vertices and edges)
Type Convex regular 4-polytope
Schläfli symbol {3,3,4}
{3,31,1}
h{4,3,3}
s{2,2,2}
Coxeter-Dynkin diagram



Cells 16 {3,3}
Faces 32 {3}
Edges 24
Vertices 8
Vertex figure
Octahedron
Petrie polygon octagon
Coxeter group C4, [3,3,4]
D4, [31,1,1]
[23] (half)
Symmetry group [3,3,4], order 384
[31,1,1], order 192
[3,4,2+], order 48
[23]+, order 8
Dual Tesseract
Properties convex, isogonal, isotoxal, isohedral
Uniform index 12

In four dimensional geometry, a 16-cell or hexadecachoron 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 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.

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 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.

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 h0,1{2,4,3}, and Coxeter diagram: .

It can also be seen as a snub 4-orthotope, represented by s{2,2,2}, and Coxeter diagram: .

Images


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.
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]
Orthogonal projection graphs

demitesseract in order-4 Petrie polygon symmetry as an alternated tesseract
Tesseract

Tessellations

One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the hexadecachoric honeycomb and has Schläfli symbol {3,3,4,3}. The dual tessellation, icositetrachoric 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 an octahedron, 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.

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:

Related uniform polytopes

Name tesseract rectified
tesseract		 truncated
tesseract		 cantellated
tesseract		 runcinated
tesseract		 bitruncated
tesseract		 cantitruncated
tesseract		 runcitruncated
tesseract		 omnitruncated
tesseract
Coxeter-Dynkin
diagram
Schläfli
symbol		 {4,3,3} t1{4,3,3} t0,1{4,3,3} t0,2{4,3,3} t0,3{4,3,3} t1,2{4,3,3} t0,1,2{4,3,3} t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4 Coxeter plane graph
 
Name 16-cell rectified
16-cell		 truncated
16-cell		 cantellated
16-cell		 runcinated
16-cell		 bitruncated
16-cell		 cantitruncated
16-cell		 runcitruncated
16-cell		 omnitruncated
16-cell
Coxeter-Dynkin
diagram
Schläfli
symbol		 {3,3,4} t1{3,3,4} t0,1{3,3,4} t0,2{3,3,4} t0,3{3,3,4} t1,2{3,3,4} t0,1,2{3,3,4} t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4 Coxeter plane graph

See also

5-cell 8-cell 16-cell 24-cell 120-cell 600-cell

{3,3,3}
pentachoron

{4,3,3}
tesseract

{3,3,4}
hexadecachoron

{3,4,3}
icositetrachoron

{5,3,3}
hecatonicosachoron

{3,3,5}
hexacosichoron

References

External links