24cell
24cell  

Schlegel diagram
(vertices and edges)  
Type  Convex regular 4polytope 
Schläfli symbol  {3,4,3} r{3,3,4} = {3^{1,1,1}} = 
Coxeter diagram 
or or 
Cells  24 {3,4} 
Faces  96 {3} 
Edges  96 
Vertices  24 
Vertex figure  Cube 
Petrie polygon  dodecagon 
Coxeter group  F_{4}, [3,4,3], order 1152 B_{4}, [4,3,3], order 384 D_{4}, [3^{1,1,1}], order 192 
Dual  Selfdual 
Properties  convex, isogonal, isotoxal, isohedral 
Uniform index  22 
In geometry, the 24cell is the convex regular 4polytope (fourdimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C_{24}, icositetrachoron^{[1]}, octaplex (short for "octahedral complex"), icosatetrahedroid,^{[2]} octacube,^{[citation needed]} hyperdiamond or polyoctahedron, being constructed of octahedral cells.
The boundary of the 24cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24cell is selfdual. In fact, the 24cell is the unique convex selfdual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analogue in 3 dimensions.
Contents
Constructions
The 24cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of
 .
Those coordinates can be constructed as , rectifying the 16cell, , with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16cell is the octahedron; thus, cutting the vertices of the 16cell at the midpoint of its incident edges produce 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16cell which become 16 octahedra, giving the 24cell 24 octahedral cells.
In this form the 24cell has edges of length √2 and is inscribed in a 3sphere of radius √2.
Remarkably, the 24cell is self dual, having the same number of vertices (24) with its cells and the same number of edges (96) with its faces.
If the dual is taken instead, and its edge length set to 1, the coordinates of another 24cell can be found which reveal more structure. In this form the vertices of the 24cell can be given as follows: 8 vertices obtained by permuting
 (±1, 0, 0, 0)
and 16 vertices of the form
 (±1/2, ±1/2, ±1/2, ±1/2).
Viewed as quaternions, these are the unit Hurwitz quaternions. Furthermore, the first 8 vertices are the vertices of a regular 16cell and the other 16 are the vertices of its dual, the tesseract. This gives a construction equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. This is equivalent to the dual of a rectified 16cell. The analogous construction in 3space gives the rhombic dodecahedron which, however, is not regular.
We can further divide the last 16 vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16cell. This shows that the vertices of the 24cell can be grouped into three disjoint sets of eight with each set defining a regular 16cell, and with the complement defining the dual tesseract. This also shows that the symmetries of the 16cell form a subgroup of index 3 of the symmetry group of the 24cell.
As a configuration
The elements of a regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (fvectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^{[3]}^{[4]}
Tessellations
A regular tessellation of 4dimensional Euclidean space exists with 24cells, called an icositetrachoric honeycomb, with Schläfli symbol {3,4,3,3}. Hence, the dihedral angle of a 24cell is 120°.^{[5]} The regular dual tessellation, {3,3,4,3} has 16cells. (See also List of regular polytopes which includes a third regular tessellation, the tesseractic honeycomb {4,3,3,4}.)
Symmetries, root systems, and tessellations
The 24 root vectors of the D_{4} root system of the simple Lie group SO(8) form the vertices of a 24cell. The vertices can be seen in 3 hyperplanes, with the 6 vertices of an octahedron cell on each of the outer hyperplanes and 12 vertices of a cuboctahedron on a central hyperplane. These vertices, combined with the 8 vertices of the 16cell, represent the 32 root vectors of the B_{4} and C_{4} simple Lie groups.
The 48 vertices (or strictly speaking their radius vectors) of the union of the 24cell and its dual form the root system of type F_{4}. The 24 vertices of the original 24cell form a root system of type D_{4}; its size has the ratio √2:1. This is likewise true for the 24 vertices of its dual. The full symmetry group of the 24cell is the Weyl group of F_{4}, which is generated by reflections through the hyperplanes orthogonal to the F_{4} roots. This is a solvable group of order 1152. The rotational symmetry group of the 24cell is of order 576.
Quaternionic interpretation
When interpreted as the quaternions, the F_{4} root lattice (which is the integral span of the vertices of the 24cell) is closed under multiplication and is therefore a ring. This is the ring of Hurwitz integral quaternions. The vertices of the 24cell form the group of units (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the binary tetrahedral group). The vertices of the 24cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24cell are those with norm squared 2. The D_{4} root lattice is the dual of the F_{4} and is given by the subring of Hurwitz quaternions with even norm squared.
Vertices of other convex regular 4polytopes also form multiplicative groups of quaternions, but few of them generate a root lattice.
Voronoi cells
The Voronoi cells of the D_{4} root lattice are regular 24cells. The corresponding Voronoi tessellation gives a tessellation of 4dimensional Euclidean space by regular 24cells. The 24cells are centered at the D_{4} lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F_{4} lattice points with odd norm squared. Each 24cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 32 neighbors with which it shares only a single vertex. Eight 24cells meet at any given vertex in this tessellation. The Schläfli symbol for this tessellation is {3,4,3,3}. The dual tessellation, {3,3,4,3}, is one by regular 16cells. Together with the regular tesseract tessellation, {4,3,3,4}, these are the only regular tessellations of R^{4}.
The unit balls inscribed in the 24cells of the above tessellation give rise to the densest lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24cell has also been shown to give the highest possible kissing number in 4 dimensions.
Projections
The vertexfirst parallel projection of the 24cell into 3dimensional space has a rhombic dodecahedral envelope. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.
The cellfirst parallel projection of the 24cell into 3dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the Waxis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.
The edgefirst parallel projection has an elongated hexagonal dipyramidal envelope, and the facefirst parallel projection has a nonuniform hexagonal biantiprismic envelope.
The vertexfirst perspective projection of the 24cell into 3dimensional space has a tetrakis hexahedral envelope. The layout of cells in this image is similar to the image under parallel projection.
The following sequence of images shows the structure of the cellfirst perspective projection of the 24cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertexcenter radius of the 24cell.
Stereographic projection 

Animated crosssection of 24cell 
A 3D projection of a 24cell performing a simple rotation. 

A stereoscopic 3D projection of an icositetrachoron (24cell). 

Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell 
Orthogonal projections
Coxeter plane  F_{4}  

Graph  
Dihedral symmetry  [12]  
Coxeter plane  B_{3} / A_{2} (a)  B_{3} / A_{2} (b) 
Graph  
Dihedral symmetry  [6]  [6] 
Coxeter plane  B_{4}  B_{2} / A_{3} 
Graph  
Dihedral symmetry  [8]  [4] 
Three Coxeter group constructions
There are two lower symmetry forms of the 24cell, derived as a rectified 16cell, with B_{4} or [3,3,4] symmetry drawn bicolored with 8 and 16 octahedral cells. Lastly it can be constructed from D_{4} or [3^{1,1,1}] symmetry, and drawn tricolored with 8 octahedra each.
Three nets of the 24cell with cells colored by D_{4}, B_{4}, and F_{4} symmetry  

Rectified demitesseract  Rectified 16cell  Regular 24cell  
D_{4}, [3^{1,1,1}], order 192  B_{4}, [3,3,4], order 384  F_{4}, [3,4,3], order 1152  
Three sets of 8 rectified tetrahedral cells  One set of 16 rectified tetrahedral cells and one set of 8 octahedral cells.  One set of 24 octahedral cells  
Vertex figure (Each edge corresponds to one triangular face, colored by symmetry arrangement) 

Visualization
The 24cell consists of 24 octahedral cells. For visualization purposes, it is convenient that the octahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 120cell). One can stack octahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 6 cells. The cell locations lend themselves to a hyperspherical description. Pick an arbitrary cell and label it the "North Pole". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "South Pole" cell. This skeleton accounts for 18 of the 24 cells (2 + 8×2). See the table below.
There is another related great circle in the 24cell, the dual of the one above. A path that traverses 6 vertices solely along edges, resides in the dual of this polytope, which is itself since it is self dual. One can easily follow this path in a rendering of the equatorial cuboctahedron crosssection.
Starting at the North Pole, we can build up the 24cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2sphere, with the equator being a great 2sphere. The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight nonmeridian and pole cells has the same relative position to each other as the cells in a tesseract (8cell), although they touch at their vertices instead of their faces.
Layer #  Number of Cells  Description  Colatitude  Region 

1  1 cell  North Pole  0°  Northern Hemisphere 
2  8 cells  First layer of meridian cells  60°  
3  6 cells  Nonmeridian / interstitial  90°  Equator 
4  8 cells  Second layer of meridian cells  120°  Southern Hemisphere 
5  1 cell  South Pole  180°  
Total  24 cells 
The 24cell can be partitioned into disjoint sets of four of these 6cell great circle rings, forming a discrete Hopf fibration of four interlocking rings. One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.
Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180  360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24cells in a plane and form a 4D honeycomb of 24cells as described previously.
One can also follow a great circle route, through the octahedrons' opposing vertices, that is four cells long. This corresponds to traversing diagonally through the squares in the cuboctahedron crosssection. The 24cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 nonmeridian (equatorial) and pole cells. The 24cell can be equipartitioned into three 8cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration.
Related complex polygons
The regular complex polygon _{4}{3}_{4}, or contains the 24 vertices of the 24cell, and 24 4edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is _{4}[3]_{4}, order 96.^{[6]}
The regular complex polytope _{3}{4}_{3}, or , in has a real representation as a 24cell in 4dimensional space. _{3}{4}_{3} has 24 vertices, and 24 3edges. Its symmetry is _{3}[4]_{3}, order 72.
The regular complex polygon _{3}{6}_{2}, or contains the 24 vertices of the 24cell. It has 16 3edges. Its symmetry is _{3}[6]_{2}, order 48.
Related 4polytopes
Several uniform 4polytopes can be derived from the 24cell via truncation:
 truncating at 1/3 of the edge length yields the truncated 24cell;
 truncating at 1/2 of the edge length yields the rectified 24cell;
 and truncating at half the depth to the dual 24cell yields the bitruncated 24cell, which is celltransitive.
The 96 edges of the 24cell can be partitioned into the golden ratio to produce the 96 vertices of the snub 24cell. This is done by first placing vectors along the 24cell's edges such that each twodimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an octahedron produces an icosahedron, or "snub octahedron."
The 24cell is the unique convex selfdual regular Euclidean polytope that is neither a polygon nor a simplex. Relaxing the condition of convexity admits two further figures: the great 120cell and grand stellated 120cell.
Related uniform polytopes
D_{4} uniform polychora  










{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} 
24cell family polytopes  

Name  24cell  truncated 24cell  snub 24cell  rectified 24cell  cantellated 24cell  bitruncated 24cell  cantitruncated 24cell  runcinated 24cell  runcitruncated 24cell  omnitruncated 24cell  
Schläfli symbol 
{3,4,3}  t_{0,1}{3,4,3} t{3,4,3} 
s{3,4,3}  t_{1}{3,4,3} r{3,4,3} 
t_{0,2}{3,4,3} rr{3,4,3} 
t_{1,2}{3,4,3} 2t{3,4,3} 
t_{0,1,2}{3,4,3} tr{3,4,3} 
t_{0,3}{3,4,3}  t_{0,1,3}{3,4,3}  t_{0,1,2,3}{3,4,3}  
Coxeter diagram 

Schlegel diagram 

F_{4}  
B_{4}  
B_{3}(a)  
B_{3}(b)  
B_{2} 
The 24cell can also be derived as a rectified 16cell:
B4 symmetry polytopes  

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} 
{3,p,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
{3,p,3}  {3,3,3}  {3,4,3}  {3,5,3}  {3,6,3}  {3,7,3}  {3,8,3}  ... {3,∞,3}  
Image  
Cells 
{3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞} 

Vertex figure 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 
See also
References
 ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
 ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
 ^ Coxeter, Complex Regular Polytopes, p.117
 ^ Coxeter, Regular polygons, p.293
 ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991).

H.S.M. Coxeter:
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973

Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995,
ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]

Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
 Weisstein, Eric W. "24Cell". MathWorld. (also under Icositetrachoron)
 Klitzing, Richard. "4D uniform polytopes (polychora) x3o4o3o  ico".

Olshevsky, George. "Icositetrachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 [2] [3,4,3]: Icositetrachoron (22)
 [3] [4,3,3]: Rectified 16cell (22)
 [4] [3^{1,1,1}]: Icositetrachoron (22)
 Der 24Zeller (24cell) Marco Möller's Regular polytopes in R^{4} (German)
External links
 Weisstein, Eric W. "24Cell". MathWorld.
 24cell animations
 24cell in stereographic projections
 24cell description and diagrams
 Petrie dodecagons in the 24cell: mathematics and animation software