# 120-cell

In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron[1] and hecatonicosahedroid.[2]

120-cell
Schlegel diagram
(vertices and edges)
TypeConvex regular 4-polytope
Schläfli symbol{5,3,3}
Coxeter diagram
Cells120 {5,3}
Faces720 {5}
Edges1200
Vertices600
Vertex figure
tetrahedron
Petrie polygon30-gon
Coxeter groupH4, [3,3,5]
Dual600-cell
Propertiesconvex, isogonal, isotoxal, isohedral

There is a newer version of this article in Wikipedia. Please read and edit the 120-cell 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:120-cell 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:120-cell. Those portions are no longer being maintained at Wikiversity and may be out of date in this version.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:120-cell, remain incomplete and under development here for future inclusion in W:120-cell.

The boundary of the 120-cell 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 4-dimensional 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 600-cell.

## Geometry

The 120-cell 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 4-polytope,[b] it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5-cell, which is not found in any of the others.[4] The 120-cell is a four-dimensional Swiss Army knife: it contains one of everything.

It is daunting but instructive to study the 120-cell, 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 4-polytopes and the history of mathematics[5] of more than 3 dimensions The Story of the 120-cell.[6]

Regular convex 4-polytopes
Symmetry group A4 B4 F4 H4
Name 5-cell

Hyper-tetrahedron
5-point

16-cell

Hyper-octahedron
8-point

8-cell

Hyper-cube
16-point

24-cell

24-point

600-cell

Hyper-icosahedron
120-point

120-cell

Hyper-dodecahedron
600-point

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 5-tetrahedron 2 8-tetrahedron 2 4-cube 4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed 120 in 120-cell 675 in 120-cell 2 16-cells 3 8-cells 25 24-cells 10 600-cells
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 30-gons 20 30-gons
Clifford polygons 3 5/2 pentagrams 6 8𝝅 4 octagrams 12 3 octagrams 16 4𝝅 3 hexagrams 40 5𝝅 1 15-grams 240 5/2 15-grams
Long radius ${\displaystyle 1}$  ${\displaystyle 1}$  ${\displaystyle 1}$  ${\displaystyle 1}$  ${\displaystyle 1}$  ${\displaystyle 1}$
Edge length ${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$  ${\displaystyle {\sqrt {2}}\approx 1.414}$  ${\displaystyle 1}$  ${\displaystyle 1}$  ${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$  ${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius ${\displaystyle {\tfrac {1}{4}}}$  ${\displaystyle {\tfrac {1}{2}}}$  ${\displaystyle {\tfrac {1}{2}}}$  ${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$  ${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$  ${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area ${\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}$  ${\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}$  ${\displaystyle 24}$  ${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$  ${\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}$  ${\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}$
Volume ${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$  ${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$  ${\displaystyle 8}$  ${\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}$  ${\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}$  ${\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content ${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$  ${\displaystyle {\tfrac {2}{3}}\approx 0.667}$  ${\displaystyle 1}$  ${\displaystyle 2}$  ${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$  ${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

### Coordinates

Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen.

The 120-cell with long radius 8 = 22 ≈ 2.828 has edge length 2/φ2 = 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}) 24-point 24-cell 600-point 120-cell ({±1, ±1, ±1, ±√5}) ({±φ−2, ±φ, ±φ, ±φ}) ({±φ−1, ±φ−1, ±φ−1, ±φ2}) ([0, ±φ−2, ±1, ±φ2]) ([0, ±φ−1, ±φ, ±√5]) ([±φ−1, ±1, ±φ, ±2])

where φ (also called τ) is the golden ratio, 1 + 5/2 ≈ 1.618.

The unit-radius 120-cell has edge length 1/φ22 ≈ 0.270.

In this frame of reference the 120-cell lies vertex up, and its coordinates[8] are the {permutations} and [even permutations] in the left column below:

 ({±1, 0, 0, 0}) 8-point 16-cell 24-point 24-cell 120-point 600-cell 600-point 120-cell ({±1/2, ±1/2, ±1/2, ±1/2}) 16-point tesseract ([±φ/2, ±1/2, ±φ−1/2, 0]) 96-point snub 24-cell ({±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}) 5-point 5-cell: ({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 unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are obtained by multiplying all possible quaternion products[9] of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the middle three columns above).[c]

The chart gives the coordinates of one instance of each of the inscribed 4-polytopes, but the 120-cell contains multiple inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.[d]

[11]

[i]

### Structure

Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter connecting each vertex to its coordinate-negation (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 unit-radius eigenvalues (chord lengths) ranging from 1/2φ2 ≈ 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 120-cell polyhedral graph are 3-colorable.

It has not been published whether the graph is Hamiltonian or Eulerian or both or neither.

#### Cell clusters

Polyhedral sections beginning with a cell.[13]

#### Geodesics

..

The 1200 5/2 ≈ 1.581 chords are the edges of the 120 disjoint 5-cells inscribed in the 120-cell.

..

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 120-cell. 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 600-cells.[k]

The sum of the squared lengths of all these distinct chords of the 120-cell is 360,000 = 6002.[l]

### Constructions

The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[b] It can be deconstructed into ten overlapping instances of its immediate predecessor (and dual) the 600-cell,[k] just as the 600-cell can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell, the 24-cell can be deconstructed into three overlapping instances of its predecessor the 8-cell (tesseract), and the 8-cell can be deconstructed into two overlapping instances of its predecessor the 16-cell.[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 600-cell's edge length is ~0.618 times its radius (the inverse golden ratio), but the 120-cell's edge length is ~0.270 times its radius.

#### Dual 600-cells

Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius (φ2/8 ≈ 0.926) and edge length of exactly 1/4. Thus the unit-edge-length 120-cell (with long radius 2φ2 ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4.

Reciprocally, the 120-cell whose coordinates are given above of long radius 8 = 22 ≈ 2.828 and edge length 2/φ2 = 3−5 ≈ 0.764 can be constructed just outside a 600-cell of slightly smaller long radius, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 600-cell must have long radius φ2, which is smaller than 8 in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ.

#### Cell rotations of inscribed duals

Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells 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 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells.

The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells 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 120-cell contains, among its many interior features, 120 compounds of ten tetrahedra.

All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.[n] Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells.

#### Augmentation

Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing 4-pyramids of some kind on the cells of the 600-cell. 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 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.[o] The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.

[19]

[k]

[20]

### As a configuration

This configuration matrix represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[21][22]

${\displaystyle {\begin{bmatrix}{\begin{matrix}600&4&6&4\\2&1200&3&3\\5&5&720&2\\20&30&12&120\end{matrix}}\end{bmatrix}}}$

Here is the configuration expanded with k-face elements and k-figures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.

H4         k-face fk f0 f1 f2 f3 k-fig Notes
A3         ( ) f0 600 4 6 4 {3,3} H4/A3 = 14400/24 = 600
A1A2         { } f1 2 720 3 3 {3} H4/A2A1 = 14400/6/2 = 1200
H2A1         {5} f2 5 5 1200 2 { } H4/H2A1 = 14400/10/2 = 720
H3         {5,3} f3 20 30 12 120 ( ) H4/H3 = 14400/120 = 120

## Visualization

The 120-cell 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 24-cell). 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

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 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. 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 Northern Hemisphere
2 12 cells First layer of meridional cells / "Arctic Circle" 36°
3 20 cells Non-meridian / interstitial 60°
4 12 cells Second layer of meridional cells / "Tropic of Cancer" 72°
5 30 cells Non-meridian / interstitial 90° Equator
6 12 cells Third layer of meridional cells / "Tropic of Capricorn" 108° Southern Hemisphere
7 20 cells Non-meridian / 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 120-cell can be partitioned into 12 cell-disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.[25][26] Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell 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 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell 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 120-cell, like the 3-sphere, 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

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 600-cell. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, 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 600-cell, traversing along edges in the 24-cell, forming a hexagon. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub 24-cell, or icosahedral pyramids in the 600-cell.

## Projections

### Orthogonal projections

Orthogonal projections of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by B.L.Chilton.[28]

The H3 decagonal projection shows the plane of the van Oss polygon.

Orthographic projections by Coxeter planes
H4 - F4

[30]
(Red=1)

[20]
(Red=1)

[12]
(Red=1)
H3 A2 / B3 / D4 A3 / B2

[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)

3-dimensional 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 four-dimensional 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 3-sphere.

A comparison of perspective projections from 3D to 2D is shown in analogy.

Comparison with regular dodecahedron
Projection Dodecahedron Dodecaplex
Schlegel diagram
12 pentagon faces in the plane

120 dodecahedral cells in 3-space
Stereographic projection
With transparent faces
Perspective projection
Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied:
• Nearest dodecahedron to the 4D viewpoint rendered in yellow
• The 12 dodecahedra immediately adjoining it rendered in cyan;
• The remaining dodecahedra rendered in green;
• Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image.
Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements:
• Four cells surrounding nearest vertex shown in 4 colors
• Nearest vertex shown in white (center of image where 4 cells meet)
• Remaining cells shown in transparent green
• Cells facing away from 4D viewpoint culled for clarity
A 3D projection of a 120-cell performing a simple rotation.
A 3D projection of a 120-cell performing a simple rotation (from the inside).
Animated 4D rotation

## Related 4-polytopes and honeycombs

### H4 polytopes

The 120-cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:

H4 family polytopes
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell

{5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3{5,3,3} tr{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}

600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell

{3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

### {p,3,3} polytopes

The 120-cell is similar to three regular 4-polytopes: the 5-cell {3,3,3} and tesseract {4,3,3} of Euclidean 4-space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure {3,3}:

### {5,3,p} polytopes

The 120-cell is a part of a sequence of 4-polytopes and honeycombs with dodecahedral cells:

### Davis 120-cell

The Davis 120-cell, introduced by Davis (1985), is a compact 4-dimensional hyperbolic manifold obtained by identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4-dimensional hyperbolic space.

## In popular culture

The video game Deltarune (2018) mentions the hyperdodecahedron on a poster, contrasting with various two-dimensional shapes.

## Notes

1. 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. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell 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 120-cell is the 600-point 4-polytope: sixth and last in the ascending sequence that begins with the 5-point 4-polytope.
3. To obtain all 600 coordinates without redundancy by quaternion cross-multiplication of these three 4-polytopes' coordinates, it is sufficient to include just one vertex of the 24-cell: (1/2, 1/2, 0, 0).[8]
4. Except in the case of the 120 5-cells, these are not counts of all the distinct 4-polytopes which can be found inscribed in the 120-cell, only the number of disjoint inscribed 4-polytopes which together comprise a 120-cell. For example, the 120-cell contains 675 distinct 16-cells.[10]
5. The point itself (𝜉i, 𝜂, 𝜉j) 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 𝜉i "longitudinal" wz plane; when 𝜂 = 𝜋/2 the point lies in the 𝜉j "equatorial" xy plane; and when 0 < 𝜂 < 𝜋/2 the point does not lie in either invariant plane. Thus the 𝜉i and 𝜉j coordinates number vertices of two completely orthogonal great circle polygons which do not intersect (at the point or anywhere else).
6. The angles 𝜉i and 𝜉j are angles of rotation in the two completely orthogonal invariant planes[e] which characterize rotations in 4-dimensional Euclidean space. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to 𝜋/2. The (𝜉i, 0, 𝜉j) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉i, 𝜋/2, 𝜉j) coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉i, 0 < 𝜂 < 𝜋/2, 𝜉j) describe the great circles (not "lines of latitude") which cross an equator but do not pass through the north or south pole.
7. The conversion from Hopf coordinates (𝜉i, 𝜂, 𝜉j) to unit-radius Cartesian coordinates (w, x, y, z) is:
w = cos 𝜉i sin 𝜂
x = cos 𝜉j cos 𝜂
y = sin 𝜉j cos 𝜂
z = sin 𝜉i sin 𝜂
The "Hopf north pole" (0, 0, 0) is Cartesian (0, 1, 0, 0).
The "Cartesian north pole" (1, 0, 0, 0) is Hopf (0, 𝜋/2, 0).
8. The Hopf coordinates (also known as the toroidal coordinates of S3) are triples of three angles:
(𝜉i, 𝜂, 𝜉j)
that parameterize the 3-sphere by numbering points along its great circles.[12] A Hopf coordinate describes a point as a rotation from the "north pole" (0, 0, 0).[f] The 𝜉i and 𝜉j coordinates range over the vertices of completely orthogonal great circle polygons which do not intersect at any vertices. Hopf coordinates are a natural alternative to Cartesian coordinates[g] for framing regular convex 4-polytopes, because the group of rotations in 4-dimensional Euclidean space, denoted SO(4), generates those polytopes. A rotation in 4D of a point {ξi, η, ξj} through angles ξ1 and ξ2 is simply expressed in Hopf coordinates as {ξi + ξ1, η, ξj + ξ2}.
9. Hopf spherical coordinates[h] of the vertices are given as three independently permuted coordinates:
(𝜉i, 𝜂, 𝜉j)𝑚
where {<k} is the {permutation} of the k non-negative integers less than k, and {≤k} is the permutation of the k+1 non-negative integers less than or equal to k. Each coordinate permutes one set of the 4-polytope's great circle polygons, so the permuted coordinate set expresses one set of rotations in 4-space which generates the 4-polytope. With Cartesian coordinates the choice of radius is a parameter determining the reference frame, but Hopf coordinates are radius-independent: all Hopf coordinates convert to unit-radius Cartesian coordinates by the same mapping. [g] Unlike Cartesian coordinates, Hopf coordinates are not necessarily unique to each point; there may be Hopf coordinate synonyms for a vertex. The multiplicity 𝑚 of the coordinate permutation is the ratio of the number of Hopf coordinates to the number of vertices.
10. The 600 vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 axes or rays of the 120-cell. 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. The vertices of the 120-cell can be partitioned into those of five disjoint 600-cells in two different ways.[17]
12. The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.[15]
13. In the dodecahedral cell of the unit-radius 120-cell, the dodecahedron (120-cell) edge length is 1/2φ2 ≈ 0.270. The orange vertices lie at the Cartesian coordinates (±8φ3, ±8φ3, ±8φ3) 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 120-cell). 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 (600-cell edges, and the 2nd chord of the 120-cell). The diameter of the dodecahedron is 3/2φ ≈ 0.757 (the cube diagonal, and the 4th chord of the 120-cell).
14. The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the ten must belong to a different 600-cell.
15. As we saw in the 600-cell, these 12 tetrahedra belong (in pairs) to the 6 icosahedral clusters of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.

## Citations

1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
2. Matila Ghyka, The Geometry of Art and Life (1977), p.68
3. Coxeter 1973, p. 293.
4. Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.
5. Mathematics and Its History, John Stillwell, 1989, 3rd edition 2010, ISBN 0-387-95336-1
6. Coxeter 1973, pp. 156-157, §8.7 Cartesian coordinates.
7. Mamone, Pileio & Levitt 2010, p. 1442, Table 3.
8. Mamone, Pileio & Levitt 2010, p. 1433, §4.1.
9. Waegell & Aravind 2014, p. 3, §2 Geometry of the 120-cell: rays and bases.
10. Zamboj 2021, pp. 10-11, §Hopf coordinates.
11. Sadoc 2001, pp. 575-576, §2.2 The Hopf fibration of S3.
12. Waegell & Aravind 2014, pp. 3-7, §2 Geometry of the 120-cell: rays and bases.
13. Copher 2019, p. 6, §3.2 Theorem 3.4.
14. Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
15. Waegell & Arvind 2014, pp. 5-6.
16. 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 mirror-image) is said to be chiral."
17. Mamone, Pileio & Levitt 2010, pp. 1438-1439, §4.5 Regular Convex 4-Polytopes; the 120-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨4.
18. Sullivan 1991, pp. 10-14, Generating the 120-cell.
19. Coxeter, Regular Polytopes, sec 1.8 Configurations
20. Coxeter, Complex Regular Polytopes, p.117
21. Sullivan 1991, p. 15, Other Properties of the 120-cell.
22. Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.
23. Zamboj 2021, pp. 6-12, §2 Mathematical background.
24. Schleimer & Segerman 2013, pp. 16-18, §6.2. Rings of dodecahedra.
25. Zamboj 2021, pp. 23-29, §5 Hopf tori corresponding to circles on B2.