In geometry, the 600cell (or hexacosichoron) is the convex regular 4polytope, or polychoron, with Schläfli symbol {3,3,5}. Its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons. Each vertex of the 600cell is a vertex of six such decagons.
The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = , 60°= , 72° = , 90° = , 108° = , 120° = , 144° = , and 180° = . Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V. References: S.L. van Oss (1899); F. Buekenhout and M. Parker (1998).
The 600cell is regarded as the 4dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex (abbreviated from “tetrahedral complex”) and polytetrahedron, being bounded by tetrahedral cells.
Its vertex figure is an icosahedron, and its dual polytope is the 120cell.
Each cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.
Coordinates
The vertices of a 600cell centered at the origin of 4space, with edges of length 1/φ (where φ = (1+√5) /2 is the golden ratio), can be given as follows: 16 vertices of the form:
 (±½,±½,±½,±½),
and 8 vertices obtained from
 (0,0,0,±1)
by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of
 ½(±φ,±1,±1/φ,0).
Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16cell, and that all 24 vertices together are vertices of a 24cell. The final 96 vertices are the vertices of a snub 24cell, which can be found by partitioning each of the 96 edges of another 24cell (dual to the first) in the golden ratio in a consistent manner.
When interpreted as quaternions, the 120 vertices of the 600cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600cell as an invariant subgroup, namely as the subgroup 2I_{L} of quaternion leftmultiplications and as the subgroup 2I_{R} of quaternion rightmultiplications. Each rotational symmetry of the 600cell is generated by specific elements of 2I_{L} and 2I_{R}; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the nonrotation Id and the central inversion Id. We have the isomorphism RSG ≅ (2I_{L} × 2I_{R}) / {Id, Id}. The order of RSG equals 120 × 120 / 2 = 7200.
The binary icosahedral group is isomorphic to SL(2,5).
The full symmetry group of the 600cell is the Weyl group of H_{4}. This is a group of order 14400. It consists of 7200 rotations and 7200 rotationreflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss (1899); see References.
Visualization
The symmetries of the 3D surface of the 600cell are somewhat difficult to visualize due to both the large number of tetrahedral cells, and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600cell is the dual of the 120cell.
Union of two tori
The 120cell can be decomposed into two disjoint tori. Since it is the dual of the 600cell, this same dual tori structure exists in the 600cell, although it is somewhat more complex. The 10cell geodesic path in the 120cell corresponds to a 10vertex decagon path in the 600cell. Start by assembling five tetrahedrons around a common edge. This structure looks somewhat like an angular “flying saucer”. Stack ten of these, vertex to vertex, “pancake” style. Fill in the annular ring between each “saucer” with 10 tetrahedrons forming an icosahedron. You can view this as five, vertex stacked, icosahedra, with the five extra annular ring gaps also filled in. The surface is the same as that of ten stacked pentagonal antiprisms. You now have a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces, 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges. The valleys are 10 edge long closed paths and correspond to other instances of the 10vertex decagon path mentioned above. These paths spiral around the center core path, but mathematically they are all equivalent. Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a clifford torus. They can be “unrolled” into square 10×10 array. See figure below.
There are exactly 50 “egg crate” recesses and peaks on both sides that mate with the 250 cell tori.
The 600cell can be further partitioned into 20 disjoint intertwining rings of 30 cells and ten edges long each, forming a discrete Hopf fibration. These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix. Five such helices nest and spiral around each of the 10vertex decagon paths, forming the initial 150 cell torus mentioned above.
This decomposition of the 600cell has symmetry [[10,2^{+},10]], order 400, the same symmetry as the grand antiprism. The grand antiprism is just the 600cell with the two above 150cell tori removed, leaving only the single middle layer of tetrahedra, similar to the belt of an icosahedron with the 5 top and 5 bottom triangles removed (pentagonal antiprism).
Images
2D projections
H_{4}  –  F_{4} 

H_{3}  A_{2} / B_{3} / D_{4}  A_{3} / B_{2} 
[10] 
[6] 
[4] 
3D projections
Vertexfirst projection  

This image shows a vertexfirst perspective projection of the 600cell into 3D. The 600cell is scaled to a vertexcenter radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:


Cellfirst projection  
This image shows the 600cell in cellfirst perspective projection into 3D. Again, the 600cell to a vertexcenter radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image. 

Stereographic projection (on 3sphere)  
CellCentered  
Simple Rotation  
A 3D projection of a 600cell performing a simple rotation. 
Frame synchronized animated comparison of the 600 cell using orthogonal isometric (left) and perspective (right) projections.
Related polytopes and honeycombs
The snub 24cell may be obtained from the 600cell by removing the vertices of an inscribed 24cell and taking the convex hull of the remaining vertices. This process is adiminishing of the 600cell.
The grand antiprism may be obtained by another diminishing of the 600cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.
120cell  rectified 120cell 
truncated 120cell 
cantellated 120cell 
runcinated 120cell 
cantitruncated 120cell 
runcitruncated 120cell 
omnitruncated 120cell 

{5,3,3}  t_{1}{5,3,3}  t_{0,1}{5,3,3}  t_{0,2}{5,3,3}  t_{0,3}{5,3,3}  t_{0,1,2}{5,3,3}  t_{0,1,3}{5,3,3}  t_{0,1,2,3}{5,3,3} 
600cell  rectified 600cell 
truncated 600cell 
cantellated 600cell 
bitruncated 600cell 
cantitruncated 600cell 
runcitruncated 600cell 
omnitruncated 600cell 
{3,3,5}  t_{1}{3,3,5}  t_{0,1}{3,3,5}  t_{0,2}{3,3,5}  t_{1,2}{3,3,5}  t_{0,1,2}{3,3,5}  t_{0,1,3}{3,3,5}  t_{0,1,2,3}{3,3,5} 
It is similar to three regular polychora: the 5cell {3,3,3}, 16cell {3,3,4} 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,∞} 
This polychora is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:
Space  S^{3}  H^{3}  

Form  Finite  Compact  Paracompact  Noncompact  
Name  {3,3,5} 
{4,3,5} 
{5,3,5} 
{6,3,5} 
{7,3,5} 
… {∞,3,5} 
Image  
Cells  {3,3} 
{4,3} 
{5,3} 
{6,3} 
{∞,3} 
{∞,3} 