In fourdimensional geometry, a 16cell or hexadecachoron is a regular convex 4polytope. It is one of the six regular convex 4polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid19th century.
It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes. The dual polytope is the tesseract (4cube).Conway’s name for a crosspolytope 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 16cell 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 16cell 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 16cell 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 44 duoprismconstruction, , of the 16cell, symmetry [[4,2^{+},4]], order 64.
Images
Stereographic projectionrowspan=2 A 3D projection of a 16cell performing a simple rotation. 
The 16cell 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] 
demitesseract in order4Petrie polygon symmetry as an alternated tesseract 
Tesseract 
Tessellations
One can tessellate 4dimensional Euclidean space by regular 16cells. 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 24cells. Together with the tesseractic honeycomb {4,3,3,4}, these are the only three regular tessellations of R^{4}. Each 16cell 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. Twentyfour 16cells meet at any given vertex in this tessellation.
Projections
The cellfirst parallel projection of the 16cell into 3space 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 (nonregular) 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 16cell, all its edges lie on the faces of the cubical envelope.
The cellfirst perspective projection of the 16cell into 3space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cellfirst parallel projection.
The vertexfirst parallel projection of the 16cell into 3space 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 16cell. The closest vertex of the 16cell to the viewer projects onto the center of the octahedron.
Finally the edgefirst parallel projection has a shortened octahedral envelope, and the facefirst parallel projection has a hexagonal bipyramidal envelope.
4 sphere Venn Diagram
The usual projection of the 16cell and 4 intersecting spheres (a Venn diagram of 4 sets) form topologically the same object in 3Dspace:
Symmetry constructions
There is a lower symmetry form of the 16cell, called a demitesseract or 4demicube, 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 4orthotope, represented by s{2^{1,1,1}}, and Coxeter diagram: .
With the tesseract constructed as a 44 duoprism, the 16cell can be seen as its dual, a 44 duopyramid.
Name  Coxeter diagram  Schläfli symbol  Coxeter notation  Order  Vertex figure 

Regular 16cell  {3,3,4}  [3,3,4]  384  
Demitesseract  = = 
h{4,3,3} {3,3^{1,1}} 
[3^{1,1,1}] = [1^{+},4,3,3]  192  
Alternated 44 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 4orthotope  s{2^{1,1,1}}  [2,2,2]^{+}  8  
4fusil  
{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
{3,3^{1,1}} h{4,3,3} 
2r{3,3^{1,1}} h_{3}{4,3,3} 
t{3,3^{1,1}} h_{2}{4,3,3} 
2t{3,3^{1,1}} h_{2,3}{4,3,3} 
r{3,3^{1,1}} {3^{1,1,1}}={3,4,3} 
rr{3,3^{1,1}} r{3^{1,1,1}}=r{3,4,3} 
tr{3,3^{1,1}} t{3^{1,1,1}}=t{3,4,3} 
sr{3,3^{1,1}} s{3^{1,1,1}}=s{3,4,3} 
The 16cell 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}  t_{1}{4,3,3} r{4,3,3} 
t_{0,1}{4,3,3} t{4,3,3} 
t_{0,2}{4,3,3} rr{4,3,3} 
t_{0,3}{4,3,3}  t_{1,2}{4,3,3} 2t{4,3,3} 
t_{0,1,2}{4,3,3} tr{4,3,3} 
t_{0,1,3}{4,3,3}  t_{0,1,2,3}{4,3,3} 
Schlegel diagram 

B_{4}  
Name  16cell  rectified 16cell 
truncated 16cell 
cantellated 16cell 
runcinated 16cell 
bitruncated 16cell 
cantitruncated 16cell 
runcitruncated 16cell 
omnitruncated 16cell 
Coxeter diagram 
= 
= 
= 
= 
= 
= 

Schläfli symbol 
{3,3,4}  t_{1}{3,3,4} r{3,3,4} 
t_{0,1}{3,3,4} t{3,3,4} 
t_{0,2}{3,3,4} rr{3,3,4} 
t_{0,3}{3,3,4}  t_{1,2}{3,3,4} 2t{3,3,4} 
t_{0,1,2}{3,3,4} tr{3,3,4} 
t_{0,1,3}{3,3,4}  t_{0,1,2,3}{3,3,4} 
Schlegel diagram 

B_{4} 
This polychoron is also related to the cubic honeycomb, order4 dodecahedral honeycomb, and order4 hexagonal tiling honeycomb all which have octahedral vertex figures.
Space  S^{3}  E^{3}  H^{3}  

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 5cell {3,3,3}, 600cell {3,3,5} of Euclidean 4space, and the order6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.
Space  S^{3}  H^{3}  

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,∞} 
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} 
Coxeter diagram 

Image  
Vertex figure r{p,3} 
Space  Euclidean 4space  Euclidean 3space  Hyperbolic 3space  

Name  {3,3,4} {3,3^{1,1}} = 
{4,3,4} {4,3^{1,1}} = 
{5,3,4} {5,3^{1,1}} = 
{6,3,4} {6,3^{1,1}} = 

Coxeter diagram 
=  =  =  =  
Image  
Cells {p,3} 