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 C16, hexadecachoron,[1] or hexdecahedroid.[2]

Regular hexadecachoron
Schlegel wireframe 16-cell.png
Schlegel diagram
(vertices and edges)
TypeConvex regular 4-polytope
Schläfli symbol{3,3,4}
Coxeter diagramCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cells16 {3,3} 3-simplex t0.svg
Faces32 {3} 2-simplex t0.svg
Vertex figure16-cell verf.png
Petrie polygonoctagon
Coxeter groupB4, [3,3,4], order 384
D4, order 192
Propertiesconvex, isogonal, isotoxal, isohedral, quasiregular
Uniform index12

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.

As a configurationEdit

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.

Orthogonal projectionsEdit

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
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 R4.

Boerdijk–Coxeter helixEdit

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.



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 DiagramEdit

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 constructionsEdit

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:         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      
Quasiregular 16-cell
[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      
        {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 polygonsEdit

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]

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.[8]

Related uniform polytopes and honeycombsEdit

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 alsoEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds


  1. ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
  2. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
  3. ^ Coxeter 1973, p. 121, §7.21. see illustration Fig 7.2B.
  4. ^ Coxeter 1973, p. 293.
  5. ^ Coxeter and Shephard, 1991, p.30 and p.47
  6. ^ Coxeter and Shephard, 1992
  7. ^ Regular Complex Polytopes, p. 108
  8. ^ Regular Complex Polytopes, p.114
  • 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.CS1 maint: ref=harv (link)
    • 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]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

External linksEdit