# 16 cell structure – Trinh Do

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.

## 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 the alternated 4-4 duoprismconstruction, , of the 16-cell, symmetry [[4,2+,4]], order 64.

## Images

 Stereographic projection|rowspan=2| 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]
 demitesseract in order-4Petrie 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 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.

## Projections

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

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{21,1,1}, and Coxeter diagram: .

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  =
=
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]+ 8
4-fusil
{3,3,4} [3,3,4] 384
{4}+{4} [[4,2,4]] 128
{3,4}+{} [4,3,2] 96
{4}+{}+{} [4,2,2] 32
{}+{}+{}+{} [2,2,2] 16

## Related uniform polytopes and honeycombs

D4 uniform polychora

{3,31,1}
h{4,3,3}
2r{3,31,1}
h3{4,3,3}
t{3,31,1}
h2{4,3,3}
2t{3,31,1}
h2,3{4,3,3}
r{3,31,1}
{31,1,1}={3,4,3}
rr{3,31,1}
r{31,1,1}=r{3,4,3}
tr{3,31,1}
t{31,1,1}=t{3,4,3}
sr{3,31,1}
s{31,1,1}=s{3,4,3}

The 16-cell is a part of the tesseractic family of uniform polychora:

Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram

Schläfli
symbol
{4,3,3} t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4
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
diagram

Schläfli
symbol
{3,3,4} t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4

This polychoron is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures.

{p,3,4}
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,4}

{4,3,4}

{5,3,4}

{6,3,4}

{7,3,4}

{8,3,4}

… {∞,3,4}

Image
Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{∞,3}

It is similar to three regular polychora: 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 a tetrahedral cells.

{3,3,p}
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3}
{3,3,4}

{3,3,5}
{3,3,6}

{3,3,7}
{3,3,8}

… {3,3,∞}

Image
Vertex
figure

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,∞}

Quasiregular polychora and honeycombs: h{4,p,q}
Space Finite Affine Compact Paracompact
Name h{4,3,3} h{4,3,4} h{4,3,5} h{4,3,6} h{4,4,3} h{4,4,4}
$left{3,{3atop3}right}$ $left{3,{3atop4}right}$ $left{3,{3atop5}right}$ $left{3,{3atop6}right}$ $left{4,{3atop4}right}$ $left{4,{4atop4}right}$
Coxeter
diagram
Image
Vertex
figure
r{p,3}

Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
Space Euclidean 4-space Euclidean 3-space Hyperbolic 3-space
Name {3,3,4}
{3,31,1} = $left{3,{3atop3}right}$
{4,3,4}
{4,31,1} = $left{4,{3atop3}right}$
{5,3,4}
{5,31,1} = $left{5,{3atop3}right}$
{6,3,4}
{6,31,1} = $left{6,{3atop3}right}$
Coxeter
diagram
=   =   =   =
Image
Cells
{p,3}

Source: Wikipedia