120-cell

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 = 2√2 ≈ 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:[8]

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

The unit-radius 120-cell has edge length 1/φ²√2 ≈ 0.270.

In this frame of reference the 120-cell lies vertex up, and its coordinates[9] are the {permutations} and [even permutations] in the left column below:

The table gives the coordinates of an instance of each of the inscribed 4-polytopes, but the 120-cell contains multiples of five 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.[lower-alpha 5]

In triacontagram {30/12}=6{5/2},
the 120 disjoint regular 5-cells of edge-length √5/2 which are inscribed in the 120-cell appear as 6 pentagrams, the Clifford polygon of the 5-cell.[lower-alpha 3] The 30 vertices comprise a Petrie polygon of the 120-cell,[lower-alpha 6] with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) Clifford parallel to the projection plane. Inscribed in the 3 decagons are 6 great pentagons[lower-alpha 7] in which the 6 pentagrams appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.

The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.[lower-alpha 3] These two additional chords give the 120-cell its characteristic isoclinic rotation,[lower-alpha 11] in addition to all the rotations of the other regular 4-polytopes which it inherits.[14]

The 120-cell's edges do not form ordinary great circles in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the 5-cell and the 8-cell tesseract, they form zig-zag Petrie polygons instead.[lower-alpha 9] The 120-cell's Petrie polygon is a triacontagon {30} zig-zag skew polygon. Successive vertices of the Petrie 30-gon lie in 5 different decagon central planes.[lower-alpha 6]

Since the 120-cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter. Since every regular convex 4-polytope is inscribed in the 120-cell, these 15 chords are the complete set of all the distinct chords in the regular 4-polytopes.

Considering the adjacency matrix of the vertices representing the polyhedral graph of the unit-radius 120-cell, the graph diameter is 15, connecting each vertex to its coordinate-negation at a Euclidean distance of 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,[lower-alpha 3] 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 1/φ²√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 120-cell polyhedral graph are 3-colorable.

The graph is Eulerian having degree 4 in every vertex. Its edge set can be decomposed into two Hamiltonian cycles.[15]

The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[lower-alpha 2] It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the 600-cell,[lower-alpha 12] just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the 24-cell,[lower-alpha 13] the 24-cell can be deconstructed into three distinct instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor the 16-cell.[18] The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.[19]

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.

Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.

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 φ²√2 ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The unit radius 120-cell (with edge-length 1/φ²√2 ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius √8/φ² ≈ 1.080.

One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.[lower-alpha 14]

Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius φ²/√8 ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given above of long radius √8 = 2√2 ≈ 2.828 and edge-length 2/φ² = 3−√5 ≈ 0.764 can be constructed in this manner just outside a 600-cell of 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. The 120-cell of edge-length 2 and long radius φ²√8 ≈ 7.405 given by Coxeter[3] can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³.

Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps.

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).[lower-alpha 12] 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.[21] Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). 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 of a cube obviously).[22] This shows that the 120-cell contains, among its many interior features, 120 compounds of ten tetrahedra, each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.

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.[lower-alpha 15] Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells.

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 four '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.[lower-alpha 16] The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.

Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits ${\displaystyle O(\Lambda )=W(H_{4})=I}$ of order 120.[23] The following describe ${\displaystyle T}$ and ${\displaystyle T'}$ 24-cells as quaternion orbit weights of D4 under the Weyl group W(D4):
O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
O(1000) : V1
O(0010) : V2
O(0001) : V3

$${\displaystyle T'={\sqrt {2}}\{V1\oplus V2\oplus V3\}={\begin{pmatrix}{\frac {-1-e_{1}}{\sqrt {2}}}&{\frac {1-e_{1}}{\sqrt {2}}}&{\frac {-1+e_{1}}{\sqrt {2}}}&{\frac {1+e_{1}}{\sqrt {2}}}&{\frac {-e_{2}-e_{3}}{\sqrt {2}}}&{\frac {e_{2}-e_{3}}{\sqrt {2}}}&{\frac {-e_{2}+e_{3}}{\sqrt {2}}}&{\frac {e_{2}+e_{3}}{\sqrt {2}}}\\{\frac {-1-e_{2}}{\sqrt {2}}}&{\frac {1-e_{2}}{\sqrt {2}}}&{\frac {-1+e_{2}}{\sqrt {2}}}&{\frac {1+e_{2}}{\sqrt {2}}}&{\frac {-e_{1}-e_{3}}{\sqrt {2}}}&{\frac {e_{1}-e_{3}}{\sqrt {2}}}&{\frac {-e_{1}+e_{3}}{\sqrt {2}}}&{\frac {e_{1}+e_{3}}{\sqrt {2}}}\\{\frac {-e_{1}-e_{2}}{\sqrt {2}}}&{\frac {e_{1}-e_{2}}{\sqrt {2}}}&{\frac {-e_{1}+e_{2}}{\sqrt {2}}}&{\frac {e_{1}+e_{2}}{\sqrt {2}}}&{\frac {-1-e_{3}}{\sqrt {2}}}&{\frac {1-e_{3}}{\sqrt {2}}}&{\frac {-1+e_{3}}{\sqrt {2}}}&{\frac {1+e_{3}}{\sqrt {2}}}\end{pmatrix}};}$$

With quaternions ${\displaystyle (p,q)}$ where ${\displaystyle {\bar {p}}}$ is the conjugate of ${\displaystyle p}$ and ${\displaystyle [p,q]:r\rightarrow r'=prq}$ and ${\displaystyle [p,q]^{*}:r\rightarrow r''=p{\bar {r}}q}$, then the Coxeter group ${\displaystyle W(H_{4})=\lbrace [p,{\bar {p}}]\oplus [p,{\bar {p}}]^{*}\rbrace }$ is the symmetry group of the 600-cell and the 120-cell of order 14400.

Given ${\displaystyle p\in T}$ such that ${\displaystyle {\bar {p}}=\pm p^{4},{\bar {p}}^{2}=\pm p^{3},{\bar {p}}^{3}=\pm p^{2},{\bar {p}}^{4}=\pm p}$ and ${\displaystyle p^{\dagger }}$ as an exchange of ${\displaystyle -1/\varphi \leftrightarrow \varphi }$ within ${\displaystyle p}$, we can construct:

• the snub 24-cell ${\displaystyle S=\sum _{i=1}^{4}\oplus p^{i}T}$
• the 600-cell ${\displaystyle I=T+S=\sum _{i=0}^{4}\oplus p^{i}T}$
• the 120-cell ${\displaystyle J=\sum _{i,j=0}^{4}\oplus p^{i}{\bar {p}}^{\dagger j}T'}$
• the alternate snub 24-cell ${\displaystyle S'=\sum _{i=1}^{4}\oplus p^{i}{\bar {p}}^{\dagger i}T'}$
• the dual snub 24-cell = ${\displaystyle T\oplus T'\oplus S'}$.

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.[24][25]

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

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

The cell locations lend themselves to a hyperspherical description.[27] 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.

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.

Two intertwining rings of the 120-cell.

Two orthogonal rings in a cell-centered projection

The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.[28][29][30][31][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.[32] 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 edges alternating with 6 cell diameter chords.[lower-alpha 18] 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. 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), and to a great hexagon or a ring of six octahedra meeting face to face in the 24-cell.[lower-alpha 17]

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.[33]

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

Orthographic projections by Coxeter planes[34]

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 orthographic projections

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 four-dimensional 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 3-sphere.

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

Comparison with regular dodecahedron

Projection	Dodecahedron	120-cell
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

The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:[35]

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}

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}:

{p,3,3} polytopes
Space		S3			H3
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 120-cell is a part of a sequence of 4-polytopes and honeycombs with dodecahedral cells:

{5,3,p} polytopes
Space	S3	H3
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,∞}

Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.[lower-alpha 19]

In the tetrahedrally diminished dodecahedron, 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.

Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the tetrahedrally diminished dodecahedron because the 4 vertices removed formed a tetrahedron inscribed in the dodecahedron. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord. The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces.

Since the vertex figure of the 120-cell is the tetrahedron, each truncated vertex is replaced by a tetrahedron,[lower-alpha 20] leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges.

The 480-point diminished 120-cell may be called the tetrahedrally diminished 120-cell because its cells are tetrahedrally diminished, or the 600-cell diminished 120-cell because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the regular 5-cells diminished 120-cell because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.[lower-alpha 3]

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.