120cell
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]}
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 
There is a newer version of this article in Wikipedia. Please read and edit the 120cell article on Wikipedia. This version was a draft of an update to the Wikipedia article of the same name. It was originally derived from W:120cell by copying that article (with all its history) to Wikiversity. After a long period of editing at Wikiversity, and substantial expansion, portions of it were merged back into W:120cell. Those portions are no longer being maintained at Wikiversity and may be out of date in this version. Other portions of this version, not yet merged back into W:120cell, remain incomplete and under development here for future inclusion in W:120cell.
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.
Geometry Edit
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  5 unigons x 3  2 squares x 3  4 rectangles x 3  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon  1 octagon  2 octagons  2 dodecagons  4 30gons  20 30gons  
Clifford polygons  3 √5/2 pentagrams  6 8𝝅 √4 octagrams  12 √3 octagrams  16 4𝝅 √3 hexagrams  40 5𝝅 √1 15grams  240 √5/2 15grams  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
Coordinates Edit
Natural Cartesian coordinates for a 4polytope centered at the origin of 4space occur in different frames of reference, depending on the long radius (centertovertex) chosen.
√8 radius Cartesian coordinates Edit
The 120cell with long radius √8 = 2√2 ≈ 2.828 has edge length 2/φ^{2} = 3−√5 ≈ 0.764.
In this frame of reference, its 600 vertex coordinates are the {permutations} and [even permutations] of the following:^{[7]}
({0, 0, ±2, ±2})  24point 24cell  600point 120cell 
({±1, ±1, ±1, ±√5})  
({±φ^{−2}, ±φ, ±φ, ±φ})  
({±φ^{−1}, ±φ^{−1}, ±φ^{−1}, ±φ^{2}})  
([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 ≈ 1.618.
Unit radius Cartesian coordinates Edit
The unitradius 120cell has edge length 1/φ^{2}√2 ≈ 0.270.
In this frame of reference the 120cell lies vertex up, and its coordinates^{[8]} are the {permutations} and [even permutations] in the left column below:
({±1, 0, 0, 0})  8point 16cell  24point 24cell  120point 600cell  600point 120cell 
({±1/2, ±1/2, ±1/2, ±1/2})  16point tesseract  
([±φ/2, ±1/2, ±φ^{−1}/2, 0])  96point snub 24cell  
({±1/√2, ±1/√2, 0, 0}) ({±φ^{−1}/√8, ±φ^{−1}/√8, ±φ^{−1}/√8, ±φ^{−1}/√8}) ({±1/√8, ±1/√8, ±1/√8, ±√5/√8}) 
5point 5cell:
({1, 0, 0, 0}) ({−1/4, √5/4, √5/4, √5/4}) ({−1/4, −√5/4, −√5/4, √5/4}) ({−1/4, −√5/4, √5/4, −√5/4}) ({−1/4, √5/4, −√5/4, −√5/4}) 
({±1/√2, ±1/√2, 0, 0})  ({±1, 0, 0, 0})
({±1/2, ±1/2, ±1/2, ±1/2}) ([±φ/2, ±1/2, ±φ^{−1}/2, 0]) 
The unitradius coordinates of the 600 vertices of the 120cell (in the left column above) are obtained by multiplying all possible quaternion products^{[9]} of the 5 vertices of the 5cell, the 24 vertices of the 24cell, and the 120 vertices of the 600cell (in the middle three columns above).^{[c]}
The chart gives the coordinates of one instance of each of the inscribed 4polytopes, but the 120cell contains multiple inscribed instances of each of its precursor 4polytopes, occupying different subsets of its vertices. The (600point) 120cell is the convex hull of 5 disjoint (120point) 600cells. Each (120point) 600cell is the convex hull of 5 disjoint (24point) 24cells, so the 120cell is the convex hull of 25 disjoint 24cells. Each 24cell is the convex hull of 3 disjoint (8point) 16cells, so the 120cell is the convex hull of 75 disjoint 16cells. Uniquely, the (600point) 120cell is the convex hull of 120 disjoint (5point) 5cells.^{[d]}
Hopf spherical coordinates Edit
^{[11]}
^{[i]}
Structure Edit
Vertex adjacency Edit
Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter connecting each vertex to its coordinatenegation (antipodal vertex) is 15, 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 unitradius eigenvalues (chord lengths) ranging from 1/√2φ^{2} ≈ 0.270 with a multiplicity of 4, to 2 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 or Eulerian or both or neither.
Cell clusters Edit
Polyhedral sections beginning with a cell.^{[13]}
Geodesics Edit
..
The 1200 √5/√2 ≈ 1.581 chords are the edges of the 120 disjoint 5cells inscribed in the 120cell.
..
The √4 chords occur as 300 long diameters (675 sets of 4 orthogonal axes, with each axis occurring in 9 sets),^{[j]} the 600 long radii of the 120cell. The √4 chords join opposite vertices which are five √0.𝚫 chords apart on a geodesic great circle. There are 10 distinct but overlapping sets of 60 diameters, each comprising one of the 10 inscribed 600cells.^{[k]}
The sum of the squared lengths of all these distinct chords of the 120cell is 360,000 = 600^{2}.^{[l]}
Constructions Edit
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 overlapping instances of its immediate predecessor (and dual) the 600cell,^{[k]} just as the 600cell can be deconstructed into twentyfive overlapping instances of its immediate predecessor the 24cell, the 24cell can be deconstructed into three overlapping instances of its predecessor the 8cell (tesseract), and the 8cell can be deconstructed into two overlapping instances of its predecessor the 16cell.^{[16]}
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.
Dual 600cells Edit
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 √2φ^{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.
Cell rotations of inscribed duals Edit
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).^{[k]} 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.^{[18]} 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). 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.^{[n]} Therefore the whole 120cell, with all ten inscribed 600cells, can be generated from just one 600cell by rotating its cells.
Augmentation Edit
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.^{[o]} The central cell is vertexbonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.
Symmetry operations Edit
^{[19]}
Rotations Edit
^{[k]}
Reflections Edit
^{[20]}
As a configuration Edit
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.^{[21]}^{[22]}
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 
Visualization Edit
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.^{[23]}
Layered stereographic projection Edit
The cell locations lend themselves to a hyperspherical description.^{[24]} 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 Edit
The 120cell can be partitioned into 12 celldisjoint 10cell great circle rings, forming a discrete/quantized Hopf fibration.^{[25]}^{[26]} 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.^{[27]} 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.
Other great circle constructs Edit
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: an irregular dodecagon. 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, traversing along edges in the 24cell, forming a hexagon. 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 Edit
Orthogonal projections Edit
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.^{[28]}
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 
Perspective projections Edit
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 as 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 4polytopes and honeycombs Edit
H_{4} polytopes Edit
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} 
{p,3,3} polytopes Edit
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} 
{5,3,p} polytopes Edit
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,∞} 
Davis 120cell Edit
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.
In popular culture Edit
The video game Deltarune (2018) mentions the hyperdodecahedron on a poster, contrasting with various twodimensional shapes.
See also Edit
 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 Edit
 ↑ 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.^{[3]}
 ↑ ^{2.0} ^{2.1} The convex regular 4polytopes can be ordered by size as a measure of 4dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more 4content within the same radius. The 4simplex (5cell) is the limit smallest case, and the 120cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 120cell is the 600point 4polytope: sixth and last in the ascending sequence that begins with the 5point 4polytope.
 ↑ To obtain all 600 coordinates without redundancy by quaternion crossmultiplication of these three 4polytopes' coordinates, it is sufficient to include just one vertex of the 24cell: (1/√2, 1/√2, 0, 0).^{[8]}
 ↑ Except in the case of the 120 5cells, these are not counts of all the distinct 4polytopes which can be found inscribed in the 120cell, only the number of disjoint inscribed 4polytopes which together comprise a 120cell. For example, the 120cell contains 675 distinct 16cells.^{[10]}
 ↑ The point itself (𝜉_{i}, 𝜂, 𝜉_{j}) does not necessarily lie in either of the invariant planes of rotation referenced to locate it (by convention, the wz and xy Cartesian planes), and never lies in both of them, since completely orthogonal planes do not intersect at any point except their common center. When 𝜂 = 0, the point lies in the 𝜉_{i} "longitudinal" wz plane; when 𝜂 = 𝜋/2 the point lies in the 𝜉_{j} "equatorial" xy plane; and when 0 < 𝜂 < 𝜋/2 the point does not lie in either invariant plane. Thus the 𝜉_{i} and 𝜉_{j} coordinates number vertices of two completely orthogonal great circle polygons which do not intersect (at the point or anywhere else).
 ↑ The angles 𝜉_{i} and 𝜉_{j} are angles of rotation in the two completely orthogonal invariant planes^{[e]} which characterize rotations in 4dimensional Euclidean space. The angle 𝜂 is the inclination of both these planes from the northsouth pole axis, where 𝜂 ranges from 0 to 𝜋/2. The (𝜉_{i}, 0, 𝜉_{j}) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_{i}, 𝜋/2, 𝜉_{j}) coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4polytope, as the equator of a 3sphere is a whole 2sphere of great circles. The other Hopf coordinates (𝜉_{i}, 0 < 𝜂 < 𝜋/2, 𝜉_{j}) describe the great circles (not "lines of latitude") which cross an equator but do not pass through the north or south pole.
 ↑ ^{7.0} ^{7.1} The conversion from Hopf coordinates (𝜉_{i}, 𝜂, 𝜉_{j}) to unitradius Cartesian coordinates (w, x, y, z) is:
 w = cos 𝜉_{i} sin 𝜂
 x = cos 𝜉_{j} cos 𝜂
 y = sin 𝜉_{j} cos 𝜂
 z = sin 𝜉_{i} sin 𝜂
The "Cartesian north pole" (1, 0, 0, 0) is Hopf (0, 𝜋/2, 0).  w = cos 𝜉_{i} sin 𝜂
 ↑ The Hopf coordinates (also known as the toroidal coordinates of S_{3}) are triples of three angles:
 (𝜉_{i}, 𝜂, 𝜉_{j})
 ↑ Hopf spherical coordinates^{[h]} of the vertices are given as three independently permuted coordinates:
 (𝜉_{i}, 𝜂, 𝜉_{j})_{𝑚}
 (𝜉_{i}, 𝜂, 𝜉_{j})_{𝑚}
 ↑ The 600 vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 axes or rays of the 120cell. Any set of four mutually orthogonal axes is called a basis (as in the basis for a coordinate system). The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays.^{[14]}
 ↑ ^{11.0} ^{11.1} ^{11.2} ^{11.3} The vertices of the 120cell can be partitioned into those of five disjoint 600cells in two different ways.^{[17]}
 ↑ The sum of the squared lengths of all the distinct chords of any regular convex npolytope of unit radius is the square of the number of vertices.^{[15]}
 ↑ In the dodecahedral cell of the unitradius 120cell, the dodecahedron (120cell) edge length is 1/√2φ^{2} ≈ 0.270. The orange vertices lie at the Cartesian coordinates (±√8φ^{3}, ±√8φ^{3}, ±√8φ^{3}) relative to origin at the cell center. They form a cube (dashed lines) of edge length 1/√2φ ≈ 0.437 (the pentagon diagonal, and the 1st chord of the 120cell). The face diagonals of the cube (not shown) of edge length 1/φ ≈ 0.618 are the edges of tetrahedral cells inscribed in the cube (600cell edges, and the 2nd chord of the 120cell). The diameter of the dodecahedron is √3/√2φ ≈ 0.757 (the cube diagonal, and the 4th chord of the 120cell).
 ↑ The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600cell do not, so each of the ten must belong to a different 600cell.
 ↑ As we saw in the 600cell, these 12 tetrahedra belong (in pairs) to the 6 icosahedral clusters of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.
Citations Edit
 ↑ 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 1973, p. 293.
 ↑ Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.
 ↑ Mathematics and Its History, John Stillwell, 1989, 3rd edition 2010, ISBN 0387953361
 ↑ Stillwell 2001.
 ↑ Coxeter 1973, pp. 156157, §8.7 Cartesian coordinates.
 ↑ ^{8.0} ^{8.1} Mamone, Pileio & Levitt 2010, p. 1442, Table 3.
 ↑ Mamone, Pileio & Levitt 2010, p. 1433, §4.1.
 ↑ Waegell & Aravind 2014, p. 3, §2 Geometry of the 120cell: rays and bases.
 ↑ Zamboj 2021, pp. 1011, §Hopf coordinates.
 ↑ Sadoc 2001, pp. 575576, §2.2 The Hopf fibration of S3.
 ↑ Sullivan 1991.
 ↑ Waegell & Aravind 2014, pp. 37, §2 Geometry of the 120cell: rays and bases.
 ↑ Copher 2019, p. 6, §3.2 Theorem 3.4.
 ↑ Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
 ↑ Waegell & Arvind 2014, pp. 56.
 ↑ Coxeter et al. 1938, p. 4; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a compound of five octahedra, which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a compound of ten tetrahedra. Further, we can choose one tetrahedron from each stella octangula, so as to derive a compound of five tetrahedra, which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirrorimage) is said to be chiral."
 ↑ Mamone, Pileio & Levitt 2010, pp. 14381439, §4.5 Regular Convex 4Polytopes; the 120cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨_{4}.
 ↑ Sullivan 1991, pp. 1014, Generating the 120cell.
 ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
 ↑ Coxeter, Complex Regular Polytopes, p.117
 ↑ Sullivan 1991, p. 15, Other Properties of the 120cell.
 ↑ Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.
 ↑ Zamboj 2021, pp. 612, §2 Mathematical background.
 ↑ Schleimer & Segerman 2013, pp. 1618, §6.2. Rings of dodecahedra.
 ↑ Zamboj 2021, pp. 2329, §5 Hopf tori corresponding to circles on B^{2}.
 ↑ Chilton 1963.
References Edit
 Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
 Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.
 Coxeter, H.S.M. (1995). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C. et al.. eds. Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.). WileyInterscience Publication. ISBN 9780471010036. https://www.wiley.com/enus/Kaleidoscopes%3A+Selected+Writings+of+H+S+M+Coxeterp9780471010036.
 (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]
 Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The FiftyNine Icosahedra. 6. University of Toronto Studies (Mathematical Series).
 Stillwell, John (January 2001). "The Story of the 120Cell". Notices of the AMS 48 (1): 1725. https://www.ams.org/notices/200101/feastillwell.pdf.
 J.H. Conway and M.J.T. Guy: FourDimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
 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 [1]
 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
 Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry 20 (3): 433444.
 Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494.
 Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths". arXiv:1903.06971 [math.MG].
 Miyazaki, Koji (1990). "Primary Hypergeodesic Polytopes". International Journal of Space Structures 5 (34): 309323. doi:10.1177/026635119000500312.
 van Ittersum, Clara (2020). "Symmetry groups of regular polytopes in three and four dimensions". TUDelft.
 Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry 2: 14231449. doi:10.3390/sym2031423.
 Sullivan, John M. (1991). "Generating and Rendering FourDimensional Polytopes". Mathematica Journal 1 (3).
 Waegell, Mordecai; Aravind, P.K. (10 Sep 2014). "Parity proofs of the KochenSpecker theorem based on the 120cell". arXiv:1309.7530v3 [quantph].
 Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in the double orthogonal projection of the 4space". arXiv:2003.09236v2 [math.HO].
 Sadoc, JeanFrancois (2001). "Helices and helix packings derived from the {3,3,5} polytope". W:European Physical Journal E 5: 575–582. doi:10.1007/s101890170040. https://www.researchgate.net/publication/260046074.
 Chilton, B. L. (Nov 29, 1963). "On the projection of the regular polytope {5,3,3} into a regular triacontagon". Canadian Mathematical Bulletin 7 (3). https://www.cambridge.org/core/journals/canadianmathematicalbulletin/article/ontheprojectionoftheregularpolytope533intoaregulartriacontagon/C8AC2FBB049A0C6B5BC554513168ED6E.
 Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120cell". Notices Amer. Math. Soc. 62 (11). https://arxiv.org/pdf/1310.3549.pdf.
External links Edit
 Olshevsky, George. "Hecatonicosachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Klitzing, Richard. "4D uniform polytopes (polychora)".
 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.