120cell  

Schlegel diagram (vertices and edges)  
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 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. It can be thought of as the 4dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.
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.
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✪ The 4D Dodecahedron (120 Cell)

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✪ Half of a 120Cell
Transcription
Contents
Elements
 There are 120 cells, 720 pentagonal faces, 1200 edges, and 600 vertices.
 There are 4 dodecahedra, 6 pentagons, and 4 edges meeting at every vertex.
 There are 3 dodecahedra and 3 pentagons meeting every edge.
 The dual polytope of the 120cell is the 600cell.
 The compound formed from the 120cell and its dual is the compound of 120cell and 600cell.
 The vertex figure of the 120cell is a tetrahedron.
 The dihedral angle (angle between facet hyperplanes) of the 120cell is 144°^{[3]}
As a configuration
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.^{[4]}^{[5]}
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 
Cartesian coordinates
The 600 vertices of a 120cell with an edge length of 2/φ^{2} = 3−√5 include all permutations of:^{[6]}
 (0, 0, ±2, ±2)
 (±1, ±1, ±1, ±√5)
 (±φ^{−2}, ±φ, ±φ, ±φ)
 (±φ^{−1}, ±φ^{−1}, ±φ^{−1}, ±φ^{2})
and all even permutations of
 (0, ±φ^{−2}, ±1, ±φ^{2})
 (0, ±φ^{−1}, ±φ, ±√5)
 (±φ^{−1}, ±1, ±φ, ±2)
where φ (also called τ) is the golden ratio, 1 + √5/2.
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 2−3φ, 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.
Visualization
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.
Layered stereographic projection
The cell locations lend themselves to a hyperspherical description. 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.
Intertwining rings
The 120cell can be partitioned into 12 disjoint 10cell great circle rings, forming a discrete/quantized Hopf fibration. 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. 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 great circles.
Other great circle constructs
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 vertices 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).
Projections
Orthogonal projections
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.^{[7]}
The H3 decagonal projection shows the plane of the van Oss polygon.
H_{4}    F_{4} 

[30] 
[20] 
[12] 
H_{3}  A_{2} / B_{3} / D_{4}  A_{3} / B_{2} 
[10] 
[6] 
[4] 
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 
Perspective projections
These projections use perspective projection, from a specific view point in four dimensions, and 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 the cell as the envelope of the 3D model, and other cells are smaller seen inside it. Stereographic projection use the same approach, but are shown with curved edges, representing the polytope a tiling of a 3sphere.
A comparison of perspective projections from 3D to 2D is shown in analogy.
Projection  Dodecahedron  Dodecaplex 

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 
Related polyhedra and honeycombs
The 120cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:
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} 
It is similar to three regular 4polytopes: the 5cell {3,3,3}, tesseract {4,3,3}, of Euclidean 4space, and hexagonal tiling honeycomb of hyperbolic space. All of these have a tetrahedral vertex figure.
{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} 
This honeycomb 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,∞} 
In popular culture
The video game Deltarune (2018) mentions the hyperdodecahedron on a poster, contrasting with various twodimensional shapes.
See also
 Uniform 4polytope family with [5,3,3] symmetry
 57cell – an abstract regular 4polytope constructed from 57 hemidodecahedra.
 600cell  the dual 4polytope to the 120cell
Notes
 ^ 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, p.293
 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
 ^ Coxeter, Complex Regular Polytopes, p.117
 ^ Weisstein, Eric W. "120cell". MathWorld.
 ^ "B.+L.+Chilton"+polytopes On the projection of the regular polytope {5,3,3} into a regular triacontagon, B. L. Chilton, Nov 29, 1963.
References
 H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808.
 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 SemiRegular 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]
 J.H. Conway and M.J.T. Guy: FourDimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
 Davis, Michael W. (1985), "A hyperbolic 4manifold", Proceedings of the American Mathematical Society, 93 (2): 325–328, doi:10.2307/2044771, ISSN 00029939, MR 0770546
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 Fourdimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [2]
External links
 Weisstein, Eric W. "120Cell". MathWorld.
 Olshevsky, George. "Hecatonicosachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5x  hi".
 Der 120Zeller (120cell) Marco Möller's Regular polytopes in R^{4} (German)
 120cell explorer – A free interactive program that allows you to learn about a number of the 120cell symmetries. The 120cell is projected to 3 dimensions and then rendered using OpenGL.
 Construction of the HyperDodecahedron
 YouTube animation of the construction of the 120cell Gian Marco Todesco.
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} 