120cell  

Type  Convex regular 4polytope 
Schläfli symbol  {5,3,3} 
Coxeter diagram  
Cells  120 {5,3} 
Faces  720 {5} 
Edges  1200 
Vertices  600 
Vertex figure  tetrahedron 
Petrie polygon  30gon 
Coxeter group  H_{4}, [3,3,5] 
Dual  600cell 
Properties  convex, isogonal, isotoxal, isohedral 
Uniform index  32 
In geometry, the 120cell is the convex regular 4polytope (fourdimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C_{120}, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron^{[1]} and hecatonicosahedroid.^{[2]}
The boundary of the 120cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. Together they form 720 pentagonal faces, 1200 edges, and 600 vertices. It is the 4dimensional analogue of the regular dodecahedron, since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the dodecaplex has 120 dodecahedral facets, with 3 around each edge.^{[a]} Its dual polytope is the 600cell.
The 120cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above). As the sixth and largest regular convex 4polytope,^{[b]} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5cell, which is not found in any of the others.^{[4]} The 120cell is a fourdimensional Swiss Army knife: it contains one of everything.
It is daunting but instructive to study the 120cell, because it contains examples of every relationship among all the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit. That is why Stillwell titled his paper on the 4polytopes and the history of mathematics^{[5]} of more than 3 dimensions The Story of the 120cell.^{[6]}
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 𝝅/2 squares x 3  4 𝝅/2 rectangles x 3  4 𝝅/3 hexagons x 4  12 𝝅/5 decagons x 6  50 𝝅/15 dodecagons x 4  
Petrie polygons  1 pentagon  1 octagon  2 octagons  2 dodecagons  4 30gons  20 30gons  
Isocline polygrams  1 octagram_{3} √4  2 octagram_{3} √4  4 hexagram_{2} √3  4 30gram_{2} √1  20 30gram_{2} √1  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
The 600 vertices of a 120cell with an edge length of 2/φ^{2} = 3−√5 and a centertovertex radius of √8 = 2 √2 include all permutations of:
and all even permutations of
where φ is the golden ratio, 1 + √5/2.^{[7]}
Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter is 15, connecting each vertex to its coordinatenegation, at a Euclidean distance of 4√2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from 2/φ^{2}, with a multiplicity of 4, to 4, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582.
The vertices of the 120cell polyhedral graph are 3colorable.
It has not been published whether the graph is Hamiltonian. It is Eulerian having degree 4 in every vertex.
The 120cell is the sixth in the sequence of 6 convex regular 4polytopes (in order of size and complexity).^{[b]} It can be deconstructed into ten distinct instances (or five disjoint instances) of its immediate predecessor (and dual) the 600cell,^{[c]} just as the 600cell can be deconstructed into twentyfive distinct instances (or five disjoint instances) of its immediate predecessor the 24cell,^{[d]} the 24cell can be deconstructed into three distinct instances of its predecessor the tesseract (8cell), and the 8cell can be deconstructed into two disjoint instances of its predecessor the 16cell.^{[10]} The 120cell contains 675 distinct instances (75 disjoint instances) of the 16cell.^{[11]}
The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600cell's edge length is ~0.618 times its radius (the inverse golden ratio), but the 120cell's edge length is ~0.270 times its radius.
Since the 120cell is the dual of the 600cell, it can be constructed from the 600cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600cell of unit long radius, this results in a 120cell of slightly smaller long radius (φ^{2}/√8 ≈ 0.926) and edge length of exactly 1/4. Thus the unitedgelength 120cell (with long radius φ^{2}√2 ≈ 3.702) can be constructed in this manner just inside a 600cell of long radius 4.
Reciprocally, the 120cell whose coordinates are given above of long radius √8 = 2√2 ≈ 2.828 and edge length 2/φ^{2} = 3−√5 ≈ 0.764 can be constructed just outside a 600cell of slightly smaller long radius, by placing the center of each dodecahedral cell at one of the 120 600cell vertices. The 600cell must have long radius φ^{2}, which is smaller than √8 in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600cell, so that must be φ.
Since the 120cell contains inscribed 600cells, it contains its own dual of the same radius. The 120cell contains five disjoint 600cells (ten overlapping inscribed 600cells of which we can pick out five disjoint 600cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways).^{[c]} The vertices of each inscribed 600cell are vertices of the 120cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600cells.
The dodecahedral cells of the 120cell have tetrahedral cells of the 600cells inscribed in them.^{[13]} As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair obviously).^{[14]} This shows that the 120cell contains, among its many interior features, 120 compounds of ten tetrahedra.
All ten tetrahedra can be generated by two chiral fiveclick rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600cells inscribed in the 120cell.^{[f]} Therefore the whole 120cell, with all ten inscribed 600cells, can be generated from just one 600cell by rotating its cells.
Another consequence of the 120cell containing inscribed 600cells is that it is possible to construct it by placing 4pyramids of some kind on the cells of the 600cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into several 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron.
Only 120 tetrahedral cells of each 600cell can be inscribed in the 120cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedroninscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others facebonded around it lying only partially within the dodecahedron. The central tetrahedron is edgebonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.^{[g]} The central cell is vertexbonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.
This configuration matrix represents the 120cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[15]}^{[16]}
Here is the configuration expanded with kface elements and kfigures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.
H_{4}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  kfig  Notes  

A_{3}  ( )  f_{0}  600  4  6  4  {3,3}  H_{4}/A_{3} = 14400/24 = 600  
A_{1}A_{2}  { }  f_{1}  2  720  3  3  {3}  H_{4}/A_{2}A_{1} = 14400/6/2 = 1200  
H_{2}A_{1}  {5}  f_{2}  5  5  1200  2  { }  H_{4}/H_{2}A_{1} = 14400/10/2 = 720  
H_{3}  {5,3}  f_{3}  20  30  12  120  ( )  H_{4}/H_{3} = 14400/120 = 120 
The 120cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.^{[17]}
The cell locations lend themselves to a hyperspherical description.^{[18]} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12).
Starting at the North Pole, we can build up the 120cell in 9 latitudinal layers, with allusions to terrestrial 2sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2sphere, with the equatorial centroids lying on a great 2sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles.
Layer #  Number of Cells  Description  Colatitude  Region 

1  1 cell  North Pole  0°  Northern Hemisphere 
2  12 cells  First layer of meridional cells / "Arctic Circle"  36°  
3  20 cells  Nonmeridian / interstitial  60°  
4  12 cells  Second layer of meridional cells / "Tropic of Cancer"  72°  
5  30 cells  Nonmeridian / interstitial  90°  Equator 
6  12 cells  Third layer of meridional cells / "Tropic of Capricorn"  108°  Southern Hemisphere 
7  20 cells  Nonmeridian / interstitial  120°  
8  12 cells  Fourth layer of meridional cells / "Antarctic Circle"  144°  
9  1 cell  South Pole  180°  
Total  120 cells 
The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell.
The 120cell can be partitioned into 12 disjoint 10cell great circle rings, forming a discrete/quantized Hopf fibration.^{[19]}^{[20]} Starting with one 10cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10cell rings can be placed adjacent to the original 10cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical torsion. They are all equivalent. The spiraling is a result of the 3sphere curvature. The inner ring and the five outer rings now form a six ring, 60cell solid torus. One can continue adding 10cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120cell, like the 3sphere, is the union of these two (Clifford) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.^{[21]} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint (Clifford parallel) great circles.
There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 cells and 6 edges. Both the above great circle paths have dual great circle paths in the 600cell. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600cell, forming a decagon. The alternating cell/edge path above maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six triangular bipyramids) in the 600cell. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub 24cell (or icosahedral pyramids in the 600cell).
Orthogonal projections of the 120cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30gonal projection was made in 1963 by B. L. Chilton.^{[22]}
The H3 decagonal projection shows the plane of the van Oss polygon.
H_{4}    F_{4} 

[30] (Red=1) 
[20] (Red=1) 
[12] (Red=1) 
H_{3}  A_{2} / B_{3} / D_{4}  A_{3} / B_{2} 
[10] (Red=5, orange=10) 
[6] (Red=1, orange=3, yellow=6, lime=9, green=12) 
[4] (Red=1, orange=2, yellow=4, lime=6, green=8) 
3dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image.
3D isometric projection 
Animated 4D rotation 
These projections use perspective projection, from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. Schlegel diagrams use perspective to show fourdimensional figures, choosing a point above a specific cell, thus making that cell the envelope of the 3D model, with other cells appearing smaller inside it. Stereographic projections use the same approach, but are shown with curved edges, representing the polytope as a tiling of a 3sphere.
A comparison of perspective projections from 3D to 2D is shown in analogy.
Projection  Dodecahedron  120cell 

Schlegel diagram  12 pentagon faces in the plane 
120 dodecahedral cells in 3space 
Stereographic projection  With transparent faces 
Perspective projection  

Cellfirst perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied:
 
Vertexfirst perspective projection at 5 times the distance from center to a vertex, with these enhancements:
 
A 3D projection of a 120cell performing a simple rotation.  
A 3D projection of a 120cell performing a simple rotation (from the inside).  
Animated 4D rotation 
The 120cell is one of 15 regular and uniform polytopes with the same H_{4} symmetry [3,3,5]:^{[23]}
H_{4} family polytopes  

120cell  rectified 120cell 
truncated 120cell 
cantellated 120cell 
runcinated 120cell 
cantitruncated 120cell 
runcitruncated 120cell 
omnitruncated 120cell  
{5,3,3}  r{5,3,3}  t{5,3,3}  rr{5,3,3}  t_{0,3}{5,3,3}  tr{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}  r{3,3,5}  t{3,3,5}  rr{3,3,5}  2t{3,3,5}  tr{3,3,5}  t_{0,1,3}{3,3,5}  t_{0,1,2,3}{3,3,5} 
The 120cell is similar to three regular 4polytopes: the 5cell {3,3,3} and tesseract {4,3,3} of Euclidean 4space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure {3,3}:
{p,3,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Paracompact  Noncompact  
Name  {3,3,3}  {4,3,3}  {5,3,3}  {6,3,3}  {7,3,3}  {8,3,3}  ...{∞,3,3}  
Image  
Cells {p,3} 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 
The 120cell is a part of a sequence of 4polytopes and honeycombs with dodecahedral cells:
{5,3,p} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
Name  {5,3,3}  {5,3,4}  {5,3,5}  {5,3,6}  {5,3,7}  {5,3,8}  ... {5,3,∞} 
Image  
Vertex figure 
{3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞} 
The Davis 120cell, introduced by Davis (1985), is a compact 4dimensional hyperbolic manifold obtained by identifying opposite faces of the 120cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4dimensional hyperbolic space.