120-cell

120-cell
120-cell
	Schlegel diagram; (vertices and edges)
Type	Convex regular 4-polytope
Schläfli symbol	{5,3,3}
Coxeter diagram
Cells	120 {5,3}
Faces	720 {5}
Edges	1200
Vertices	600
Vertex figure	; tetrahedron
Petrie polygon	30-gon
Coxeter group	H4, [3,3,5]
Dual	600-cell
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	32

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]

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).^[b] As the sixth and largest regular convex 4-polytope,^[c] it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5-cell,^[d] 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.^[5] That is why Stillwell titled his paper on the 4-polytopes and the history of mathematics^[6] of more than 3 dimensions The Story of the 120-cell.^[7]

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		2 squares x 3	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons	1 pentagon x 2	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Long radius	$1$	$1$	$1$	$1$	$1$	$1$
Edge length	${\sqrt {\tfrac {5}{2}}}\approx 1.581$	${\sqrt {2}}\approx 1.414$	$1$	$1$	${\tfrac {1}{\phi }}\approx 0.618$	${\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270$
Short radius	${\tfrac {1}{4}}$	${\tfrac {1}{2}}$	${\tfrac {1}{2}}$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
Area	$10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825$	$32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713$	$24$	$96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$	$1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48$	$720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366$
Volume	$5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$	$16\left({\tfrac {1}{3}}\right)\approx 5.333$	$8$	$24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314$	$600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693$	$120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118$
4-Content	${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$	${\tfrac {2}{3}}\approx 0.667$	$1$	$2$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193$

Cartesian 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.

√8 radius coordinates

The 120-cell with long radius √8 = 2√2 ≈ 2.828 has edge length 4−2φ = 3−√5 ≈ 0.764.

In this frame of reference, its 600 vertex coordinates are the {permutations} and [even permutations] of the following:^[8]

24	({0, 0, ±2, ±2})	24-cell	600-point 120-cell
96	([0, ±φ⁻¹, ±φ, ±√5])	Snub 24-cell
120^{[disputed – discuss]}	({±φ, ±φ, ±φ, ±φ⁻²})	24 5-cells^{[disputed – discuss]}
120^{[disputed – discuss]}	({±1, ±1, ±1, ±√5})	24 5-cells^{[disputed – discuss]}
120^{[disputed – discuss]}	({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²})	24 5-cells^{[disputed – discuss]}
96	([0, ±φ⁻², ±1, ±φ²])	Snub 24-cell
24^{[disputed – discuss]}	([±φ⁻¹, ±1, ±φ, ±2])	24-cell^{[disputed – discuss]}

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

Unit radius coordinates

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:

			24	120	600
8	({±1, 0, 0, 0})	16-cell	24-cell	600-cell	120-cell
16	({±1, ±1, ±1, ±1}) / 2	Tesseract	24-cell
96	([0, ±φ⁻¹, ±1, ±φ]) / 2	Snub 24-cell
480	Diminished 120-cell: 120 (±..., ±..., ±..., ±...) 120 (±..., ±..., ±..., ±...) 120 (±..., ±..., ±..., ±...) 96 (0, ±..., ±..., ±...) 24 ({±√1/2, ±√1/2, 0, 0})	5-point 5-cell: (1, 0, 0, 0) (−1, √5, √5, √5) / 4 (−1,−√5,−√5, √5) / 4 (−1,−√5, √5,−√5) / 4 (−1, √5,−√5,−√5) / 4	24-cell: ({±√1/2, ±√1/2, 0, 0})	600-cell: ({±1, 0, 0, 0}) ({±1, ±1, ±1, ±1}) / 2 ([0, ±φ⁻¹, ±1, ±φ]) / 2
	The unit-radius coordinates of uniform convex 4-polytopes are related by quaternion multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible quaternion products^[10] 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).^[e]

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.^[j]

Chords

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.^[d] These two additional chords give the 120-cell its characteristic isoclinic rotation,^[g] in addition to all the rotations of the other regular 4-polytopes which it inherits.^[13] They also give the 120-cell a characteristic great circle polygon: an irregular great hexagon in which three 120-cell edges alternate with three 5-cell edges.^[o]

The 120-cell's edges do not form regular great circle polygons 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.^[q] The 120-cell's Petrie polygon is a triacontagon {30} zig-zag skew polygon.^[k] Successive vertices of the Petrie 30-gon lie in 3 different great decagon central planes, in the 6 great pentagons inscribed in those decagons.^[m]

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, as enumerated in 15 rows of the following table.

The first thing to notice about this table is that it has eight columns, not six: in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes, the 96-point snub 24-cell and the 480-point diminished 120-cell.^[c]

Chords of the 120-cell and its inscribed 4-polytopes
Inscribed^[15]				5-cell	16-cell	8-cell	24-cell	Snub	600-cell	Dimin	120-cell
Vertices				5	8	16	24	96	120	480	600
Edges				10^[o]	24	32	96	432	720	1200	1200^[o]
Edge chord				#8	#7	#5	#5	#3	#3^[k]	#1	#1^[k]
Isocline chord				#8^[g]	#15	#15	#10	#10	#5	#5	#8^[g]
Clifford polygon^[r]				{5/2}	{8/3}	{8/3}	{6/2}	{6/2}	{15/2}	{15/2}	{15/4}
#1	1/φ²√2	Petrie 30-gon^[p]									1 1200^[g]	4 {3,3}
#1	15.5~°	√0.𝜀^[u]	0.270~								1 1200^[g]	4 {3,3}
#2	1/φ√2	face diagonal^[w]
#2	25.2~°	√0.19~	0.437~
#3	1/φ	great decagon							10^[j] 720		3600	12 {3,5}
#3	36°	√0.𝚫	0.618~						10^[j] 720		3600	12 {3,5}
#4	√3/φ√2	cell diameter^[x]
#4	44.5~°	√0.57~	0.757~
#5		great hexagon^[y]				32	225^[j] 96	225	5^[j] 1200		6000	20 {5,3}
#5	60°	√1	1			32	225^[j] 96	225	5^[j] 1200		6000	20 {5,3}
#6	√3−φ	great pentagon^[m]							720		3600	12
#6	72°	√1.𝚫	1.175~						720		3600	12
#7		great square			675^[i] 24	675 48	72		1800	7200	9000	30 r{5,3}
#7	90°	√2	1.414~		675^[i] 24	675 48	72		1800	7200	9000	30 r{5,3}
#8	√5/√2	5-cell Petrie^[z]		120^[d] 10						720	1200^[g]	4 {3,3}
#8	104.5~°	√2.5	1.581~	120^[d] 10						720	1200^[g]	4 {3,3}
#9	φ	golden section							720		3600	12
#9	108°	√2.𝚽	1.618~						720		3600	12
#10		great triangle				32	25^[j] 96		1200		6000	20
#10	120°	√3	1.732~			32	25^[j] 96		1200		6000	20
#11		{30/11}-gram^[s]
#11	..~°	√..	..~
#12	dihedral	great pent diag							720		3600	12
#12	144°	√3.𝚽	1.902~						720		3600	12
#13		{30/13}-gram
#13	..~°	√..	..~
#14		{30/14}=2{15/7}
#14	..~°	√..	..~
#15	𝝅	diameter			75^[j] 4	8	12	48	60	240	300^[i]	1
#15	180°	√4	2		75^[j] 4	8	12	48	60	240	300^[i]	1
Squared total^[aa]				25	64	256	576		14400		360000

The annotated chord table is a bill of materials for constructing the 120-cell. Only those 15 1-polytopes occur in the 120-cell, and all of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from them.

The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the #3 chord row, the 600-cell's 72 great decagons contain 720 #3 chords in all.

The red integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of 25 disjoint 24-cells (25 * 24 vertices = 600 vertices).

The green integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains 225 distinct 24-cells which share components.

The blue integers in the right column are incidence counts of the row's chord at each vertex. For example, in the #3 chord row, 12 #3 chords converge at each of the 120-cell's 600 vertices, forming an icosahedral vertex figure {3,5}.

Relationships among interior polytopes

The 120-cell is the compound of all five of the other regular convex 4-polytopes. All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.^[b] It is a four-dimensional jigsaw puzzle in which all those polytopes are the parts.^[17] Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all those parts.

The regular 1-polytope occurs in only 15 distinct lengths in the 120-cell and all its component parts.

Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four hypercubic chords √1, √2, √3 and √4 are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of those 4 chords and all the regular parts that can be built from them.

An additional 4 of the 15 chords are required to build the 600-cell. The four golden chords are square roots of irrational fractions that are functions of the golden ratio. The 600-cell is the unique solution to the combination of those 8 chords and all the regular parts that can be built from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra.

All 15 chords are required to build the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are regular 5-cells.^[ab] The relationships between the regular 5-cell (the simplex regular 4-polytope) and the other parts of the regular 4-polytopes are manifest only in the 120-cell.

Polyhedral graph

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,^[d] 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.^[19]

Concentric hulls

Constructions

The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^[c] It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the 600-cell,^[f] just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the 24-cell,^[ac] 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.^[22] The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.^[i]

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

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.

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.

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).^[f] 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.^[25] 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).^[26] 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.^[ad] 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 four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron.^[ae]

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.^[af] The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.

Weyl orbits

Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits $O(\Lambda )=W(H_{4})=I$ of order 120.^[27] The following describe $T$ and $T'$ 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

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

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

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

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.^[28]^[29]

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

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	f_k	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.^[30]

Layered stereographic projection

The cell locations lend themselves to a hyperspherical description.^[31] 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	0°	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 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.^[32]^[33]^[34]^[35]^[30] 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.^[36] 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 edges alternating with 6 cell diameter chords.^[x] 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.^[y]

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.^[37]

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

Orthographic projections by Coxeter planes^[38]
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

Perspective projections

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

Related polyhedra and honeycombs

H₄ polytopes

The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:^[39]

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}	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}


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}	t_0,1,3{3,3,5}	t_0,1,2,3{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}:

{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}

{5,3,p} polytopes

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,∞}

Tetrahedrally diminished 120-cell

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.^[ag]

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,^[ae] each truncated vertex is replaced by a tetrahedron, 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.^[d]

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.

Notes

^ In the 120-cell, 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]
^ ^a ^b The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, except for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which does occur.
^ ^a ^b ^c 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.
^ ^a ^b ^c ^d ^e ^f ^g ^h Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,^[12] of edge-length √5/2. No regular 4-polytopes except the 5-cell and the 120-cell contain √5/2 chords (the #8 chord). The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells three different ways. Each √5/2 chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells. These chords and the 120-cell edges are the only chords between vertices in disjoint 600-cells.^[h] Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and √5/2 apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells (three different ways). Each 5-cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600-cells together in a distinct isoclinic rotation.
^ 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).^[9]
^ ^a ^b ^c The vertices of the 120-cell can be partitioned into those of five disjoint 600-cells in two different ways.^[24]
^ ^a ^b ^c ^d ^e ^f ^g
In triacontagram {30/8}=2{15/4},
2 disjoint pentadecagram isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.^[n] The chords join vertices which are 8 vertices apart on the 30-vertex circumference (but 9 zig-zag edges apart on a Petrie polygon).^[o]
The characteristic isoclinic rotation of the 120-cell takes place in the invariant planes of its 1200 edges and the completely orthogonal invariant planes of its inscribed regular 5-cells' edges.^[o] There is one distinct characteristic right (left) isoclinic rotation. The rotation's isocline chords of length √5/2 are the 1200 edges of 120 disjoint regular 5-cells inscribed in the 120-cell.^[d] 15 chords join vertices 8 vertices apart on a geodesic pentadecagram circle.^[q] Successive chords of each pentadecagram are edges of the same regular 5-cell.^[h]
^ ^a ^b ^c The 120 regular 5-cells are completely disjoint. The vertices of two disjoint 5-cells are not linked by 5-cell edges, so each pentadecagram isocline of 15 #8 chords is confined to a single 5-cell.^[g] Each pentadecagram is a circular path along 15 edges belonging to the same 10-edge 5-cell. Each 5-cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600-cells together. Additionally it has two distinct pentadecagram circuits, each of which visits each vertex three times. Each pentadecagram isocline is a distinct sequence of the 15 edges of all three distinct Petrie pentagons, but it does not include any of those 5-circuits as subsequences; it is not the concatenation of the three 5-circuits, but their interleaving. Successive vertices of each pentadecagram are vertices in completely disjoint 600-cells, as are the 5 vertices of the 5-cell. Notice that the pentadecagram isocline and its distinct isoclinic rotation is a property of the individual 5-cell, independent of the existence of a 120-cell.^[11]
^ ^a ^b ^c ^d The 120-cell has 600 vertices distributed symmetrically on the surface of a [3-sphere] in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 rays [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a basis. 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.^[23]
^ ^a ^b ^c ^d ^e ^f The 120-cell can be constructed as a compound of 5 disjoint 600-cells,^[f] or 25 disjoint 24-cells, or 75 disjoint 16-cells, or 120 disjoint 5-cells. Except in the case of the 120 5-cells,^[h] these are not counts of all the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of completely disjoint inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains 10 distinct 600-cells, 225 distinct 24-cells, and 675 distinct 16-cells.^[i]
^ ^a ^b ^c ^d ^e ^f The 120-cell and 600-cell both have 30-gon Petrie polygons.^[t] They are two different skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the 600-cell's Petrie helix does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A pair of these two helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint. Both helices twist around the 0-gon axis many times in the course of one circular orbit, forming a {30/8} polygram^[p] or a {30/11} polygram^[s] respectively.
^ In 600-cell § Decagons and pentadecagrams, see the illustration of triacontagram {30/6}=6{5}.
^ ^a ^b ^c Inscribed in the 3 Clifford parallel great decagons of each 30-tetrahedron helical Petrie polygon are 6 great pentagons^[l] in which the 6 pentagrams (regular 5-cells) 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 black and white pentadecagram isoclines both act as either a right or a left isocline in the distinct right or left isoclinic rotation.
^ ^a ^b ^c ^d ^e The invariant central plane of the 120-cell's characteristic isoclinic rotation contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length √0.𝜀 (#1 chords), and 3 inscribed regular 5-cell edges of length √5/2 (#8 chords). Each √5/2 chord is spanned by 9 zig-zag edges of a Petrie 30-gon.^[p] The 120-cell contains 400 distinct irregular great hexagons, which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways.
^ ^a ^b ^c ^d The 120-cell contains 60 distinct 30-gon Petrie polygons, and can be partitioned into 20 disjoint 30-gon Petrie polygons.^[t] The Petrie 30-gon twists around its 0-gon great circle axis 8 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 120-cell Petrie polygon can be seen to be a compound triacontagram {30/8}=2{15/4} of 30 #8 chords linking pairs of vertices that are 8 vertices apart on a geodesic.^[g] The {30/8}-gram (with its #8 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).
^ ^a ^b In a polytope with a tetrahedral vertex figure, a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path Clifford polygon is the chord set of a true geodesic great circle,^[r] circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an isocline.
^ ^a ^b The edge-path of an isocline may be called the 4-polytope's Clifford polygon, as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic Clifford displacement.
^ ^a ^b ^c
The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.
The 600-cell Petrie polygon is a helical ring which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a triacontagram {30/11} of 30 #11 chords linking pairs of vertices that are 11 vertices apart on a geodesic.^[14] The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).
^ ^a ^b The regular skew 30-gon is the Petrie polygon of the 600-cell and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell Boerdijk–Coxeter helix rings (the Petrie polygons of the 600-cell):^[s] connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell).
^ The fractional square root chord lengths are given as decimal fractions where:
𝚽 ≈ 0.618 is the inverse golden ratio 1/φ
𝚫 = 1 - 𝚽 = 𝚽² ≈ 0.382
𝜀 = √𝚫²/2 ≈ 0.073
and the 120-cell edge-length 1/φ²√2 is:
𝛇 = √𝜀 ≈ 0.270
For example:
𝛇 = √0.𝜀 = √0.073~ ≈ 0.270
^ ^a ^b ^c In the dodecahedral cell of the unit-radius 120-cell, the length of the edge (the #1 chord of the 120-cell) is 1/φ²√2 ≈ 0.270. The orange vertices lie at the Cartesian coordinates (±φ³√8, ±φ³√8, ±φ³√8) relative to origin at the cell center. They form a cube (dashed lines) of edge length 1/φ√2 ≈ 0.437 (the pentagon diagonal, and the #2 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 #3 chord of the 120-cell). The diameter of the dodecahedron is √3/φ√2 ≈ 0.757 (the cube diagonal, and the #4 chord of the 120-cell).
^ The face pentagon edge (the 120-cell edge, the #1 chord) is in the golden ratio φ ≈ 1.618 to the face pentagon diagonal (the #2 chord).^[v]
^ ^a ^b The 120-cell has a great circle polygon of 6 edges (#1 chords) alternating with 6 dodecahedron cell-diameters (#4 chords).^[v] This great circle polygon is an irregular dodecagon {12} with alternating edges of two different sizes. Two great hexagons with edges of a third size (√1, the #5 chord) are inscribed in the dodecagon.^[y] The {12} great circle is contained within a helical zig-zag of {30} edges (#1 chords) which is the Petrie polygon of the 120-cell.^[k] Hexagon vertices are 5 zig-zag edges apart. The 6 hexagon edges (#5 chords) and the 6 120-cell edges (#1 chords) of the dodecagon lie on the great circle, but the other 24 zig-zag edges (#1 chords) bridging the 6 cell diameters (#4 chords) do not lie on this great circle.
^ ^a ^b ^c
Triacontagram {30/5}=5{6}, the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.
Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a 0-gon great circle axis that does not intersect any vertices.^[k] There are 5 great hexagons inscribed in each Petrie polygon, in 5 different central planes.
^ The Petrie polygon of the 5-cell is the pentagon (5-gon), and the Petrie polygon of the 120-cell is the triacontagon (30-gon).^[p] Each 120-cell Petrie 30-gon lies completely orthogonal to six 5-cell Petrie 5-gons, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.^[d]
^ 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.^[16]
^ Dodecahedra emerge as visible features in the 120-cell, but they also occur in the 600-cell as interior polytopes.^[18]
^ In the 120-cell, each 24-cell belongs to two different 600-cells.^[20] The 120-cell can be partitioned into 25 disjoint 24-cells.^[21]
^ 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.
^ ^a ^b Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular 5-cell with a regular tetrahedron base.
^ 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.
^ The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the diminishment of the 120-point 600-cell by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point snub 24-cell. Similarly, the 8-cell tesseract can be seen as a 16-point diminished 24-cell from which one 8-point 16-cell has been removed.

Citations

^ 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
^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
^ ^a ^b Coxeter 1973, pp. 292–293, Table I(ii); "120-cell".
^ Dechant 2021, p. 18, Remark 5.7, explains why not.
^ Dechant 2021, Abstract; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."
^ Mathematics and Its History, John Stillwell, 1989, 3rd edition 2010, ISBN 0-387-95336-1
^ Stillwell 2001.
^ Coxeter 1973, pp. 156–157, §8.7 Cartesian coordinates.
^ ^a ^b Mamone, Pileio & Levitt 2010, p. 1442, Table 3.
^ Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors $\left(w,x,y,z\right)_{1}$ and $\left(w,x,y,z\right)_{2}$ according to
${\begin{pmatrix}w_{2}\\x_{2}\\y_{2}\\z_{2}\end{pmatrix}}*{\begin{pmatrix}w_{1}\\x_{1}\\y_{1}\\z_{1}\end{pmatrix}}={\begin{pmatrix}{w_{2}w_{1}-x_{2}x_{1}-y_{2}y_{1}-z_{2}z_{1}}\\{w_{2}x_{1}+x_{2}w_{1}+y_{2}z_{1}-z_{2}y_{1}}\\{w_{2}y_{1}-x_{2}z_{1}+y_{2}w_{1}+z_{2}x_{1}}\\{w_{2}z_{1}+x_{2}y_{1}-y_{2}x_{1}+z_{2}w_{1}}\end{pmatrix}}$
^ Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the pentadecagram isoclinic rotation of an individual 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the same pentadecagram isoclinic rotation of the 120-cell, its characteristic rotation. Mamone's q3 is the #8 chord of the 120-cell, the 5-cell's √5/2 edge; q13 is the #1 chord of the 120-cell, its edge.
^ Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.
^ Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄; the 120-cell has 7200 distinct simple rotations (and 7200 reflections).
^ Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries.
^ Coxeter 1973, p. 147, §8.1 The simple truncations of the general regular polytope; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."
^ Copher 2019, p. 6, §3.2 Theorem 3.4.
^ Schleimer & Segerman 2013.
^ Coxeter 1973, p. 298, Table V: (iii) Sections of {3,3,5} beginning with a vertex.
^ Carlo H. Séquin. "Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes". Retrieved March 13, 2023.
^ van Ittersum 2020, p. 435, §4.3.5 The two 600-cells circumscribing a 24-cell.
^ Denney et al. 2020, p. 5, §2 The Labeling of H4.
^ Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
^ Waegell & Aravind 2014, pp. 3–4, §2 Geometry of the 120-cell: rays and bases.
^ Waegell & Aravind 2014, pp. 5–6.
^ Sullivan 1991, pp. 4–5, The Dodecahedron.
^ 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."
^ Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell.
^ Coxeter 1973, §1.8 Configurations.
^ Coxeter 1991, p. 117.
^ ^a ^b Sullivan 1991, p. 15, Other Properties of the 120-cell.
^ Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.
^ Coxeter 1970, pp. 19–23, §9. The 120-cell and the 600-cell.
^ Schleimer & Segerman 2013, pp. 16–18, §6.2. Rings of dodecahedra.
^ Banchoff 2013.
^ Zamboj 2021, pp. 6–12, §2 Mathematical background.
^ Zamboj 2021, pp. 23–29, §5 Hopf tori corresponding to circles on B².
^ Chilton 1964.
^ Dechant 2021, pp. 18–20, 6. The Coxeter Plane.
^ Denney et al. 2020.

References

Coxeter, H.S.M. (1973) [1948]. 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.; Weiss, Asia Ivic (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.). Wiley-Interscience Publication. ISBN 978-0-471-01003-6.
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies (Mathematical Series).
Coxeter, H.S.M. (1970), "Twisted Honeycombs", Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 4, Providence, Rhode Island: American Mathematical Society
Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.
J.H. Conway and M.J.T. Guy: Four-Dimensional 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
Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [1]
Davis, Michael W. (1985), "A hyperbolic 4-manifold", Proceedings of the American Mathematical Society, 93 (2): 325–328, doi:10.2307/2044771, ISSN 0002-9939, JSTOR 2044771, 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): 433–444. arXiv:1912.06156v1. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine. 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048.
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 (3–4): 309–323. doi:10.1177/026635119000500312. S2CID 113846838.
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 (3): 1423–1449. doi:10.3390/sym2031423.
Sullivan, John M. (1991). "Generating and Rendering Four-Dimensional Polytopes". Mathematica Journal. 1 (3): 76–85.
Waegell, Mordecai; Aravind, P.K. (10 Sep 2014). "Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell". Foundations of Physics. 44 (10): 1085–1095. arXiv:1309.7530v3. doi:10.1007/s10701-014-9830-0. S2CID 254504443.
Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236v2. doi:10.1093/jcde/qwab018.
Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Chilton, B. L. (September 1964). "On the projection of the regular polytope {5,3,3} into a regular triacontagon". Canadian Mathematical Bulletin. 7 (3): 385–398. doi:10.4153/CMB-1964-037-9.
Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell". Notices Amer. Math. Soc. 62 (11): 1309–1316. arXiv:1310.3549. doi:10.1090/noti1297. S2CID 117636740.
Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.
Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)". Int. J. Geom. Methods Mod. Phys. 09 (8). arXiv:1106.3433. doi:10.1142/S0219887812500685. S2CID 119288632.
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. doi:10.1007/s00006-021-01139-2.

External links

Weisstein, Eric W. "120-Cell". MathWorld.
Olshevsky, George. "Hecatonicosachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 32, George Olshevsky.
Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5x - hi".
Der 120-Zeller (120-cell) Marco Möller's Regular polytopes in R⁴ (German)
120-cell explorer – A free interactive program that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL.
Construction of the Hyper-Dodecahedron
YouTube animation of the construction of the 120-cell Gian Marco Todesco.

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}	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}


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}	t_0,1,3{3,3,5}	t_0,1,2,3{3,3,5}

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[4] In the 120-cell, 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]

[elements-5] The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, except for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which does occur.

[polytopes_ordered_by_size_and_complexity-6] 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.

[inscribed_5-cells-7] ^ ^a ^b ^c ^d ^e ^f ^g ^h Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,^[12] of edge-length √5/2. No regular 4-polytopes except the 5-cell and the 120-cell contain √5/2 chords (the #8 chord). The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells three different ways. Each √5/2 chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells. These chords and the 120-cell edges are the only chords between vertices in disjoint 600-cells.^[h] Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and √5/2 apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells (three different ways). Each 5-cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600-cells together in a distinct isoclinic rotation.

[15] 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).^[9]

[2_ways_to_get_5_disjoint_600-cells-16] The vertices of the 120-cell can be partitioned into those of five disjoint 600-cells in two different ways.^[24]

[<span_class="nowrap">&radic;<span_style="border-top:1px_solid;_padding:0_0.1em;">5/2</span></span>_pentadecagram_isoclines-17] ^ ^a ^b ^c ^d ^e ^f ^g
In triacontagram {30/8}=2{15/4},
2 disjoint pentadecagram isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.^[n] The chords join vertices which are 8 vertices apart on the 30-vertex circumference (but 9 zig-zag edges apart on a Petrie polygon).^[o]
The characteristic isoclinic rotation of the 120-cell takes place in the invariant planes of its 1200 edges and the completely orthogonal invariant planes of its inscribed regular 5-cells' edges.^[o] There is one distinct characteristic right (left) isoclinic rotation. The rotation's isocline chords of length √5/2 are the 1200 edges of 120 disjoint regular 5-cells inscribed in the 120-cell.^[d] 15 chords join vertices 8 vertices apart on a geodesic pentadecagram circle.^[q] Successive chords of each pentadecagram are edges of the same regular 5-cell.^[h]

[five_distinct_circuits_of_the_5-cell-19] The 120 regular 5-cells are completely disjoint. The vertices of two disjoint 5-cells are not linked by 5-cell edges, so each pentadecagram isocline of 15 #8 chords is confined to a single 5-cell.^[g] Each pentadecagram is a circular path along 15 edges belonging to the same 10-edge 5-cell. Each 5-cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600-cells together. Additionally it has two distinct pentadecagram circuits, each of which visits each vertex three times. Each pentadecagram isocline is a distinct sequence of the 15 edges of all three distinct Petrie pentagons, but it does not include any of those 5-circuits as subsequences; it is not the concatenation of the three 5-circuits, but their interleaving. Successive vertices of each pentadecagram are vertices in completely disjoint 600-cells, as are the 5 vertices of the 5-cell. Notice that the pentadecagram isocline and its distinct isoclinic rotation is a property of the individual 5-cell, independent of the existence of a 120-cell.^[11]

[rays_and_bases-20] The 120-cell has 600 vertices distributed symmetrically on the surface of a [3-sphere] in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 rays [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a basis. 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.^[23]

[inscribed_counts-21] ^ ^a ^b ^c ^d ^e ^f The 120-cell can be constructed as a compound of 5 disjoint 600-cells,^[f] or 25 disjoint 24-cells, or 75 disjoint 16-cells, or 120 disjoint 5-cells. Except in the case of the 120 5-cells,^[h] these are not counts of all the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of completely disjoint inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains 10 distinct 600-cells, 225 distinct 24-cells, and 675 distinct 16-cells.^[i]

[two_different_Petrie_30-gons-22] ^ ^a ^b ^c ^d ^e ^f The 120-cell and 600-cell both have 30-gon Petrie polygons.^[t] They are two different skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the 600-cell's Petrie helix does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A pair of these two helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint. Both helices twist around the 0-gon axis many times in the course of one circular orbit, forming a {30/8} polygram^[p] or a {30/11} polygram^[s] respectively.

[23] In 600-cell § Decagons and pentadecagrams, see the illustration of triacontagram {30/6}=6{5}.

[great_pentagon-24] Inscribed in the 3 Clifford parallel great decagons of each 30-tetrahedron helical Petrie polygon are 6 great pentagons^[l] in which the 6 pentagrams (regular 5-cells) 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.

[26] The black and white pentadecagram isoclines both act as either a right or a left isocline in the distinct right or left isoclinic rotation.

[irregular_great_hexagon-27] The invariant central plane of the 120-cell's characteristic isoclinic rotation contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length √0.𝜀 (#1 chords), and 3 inscribed regular 5-cell edges of length √5/2 (#8 chords). Each √5/2 chord is spanned by 9 zig-zag edges of a Petrie 30-gon.^[p] The 120-cell contains 400 distinct irregular great hexagons, which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways.

[120-cell_Petrie_{30}-gon-28] The 120-cell contains 60 distinct 30-gon Petrie polygons, and can be partitioned into 20 disjoint 30-gon Petrie polygons.^[t] The Petrie 30-gon twists around its 0-gon great circle axis 8 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 120-cell Petrie polygon can be seen to be a compound triacontagram {30/8}=2{15/4} of 30 #8 chords linking pairs of vertices that are 8 vertices apart on a geodesic.^[g] The {30/8}-gram (with its #8 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).

[non-planar_geodesic_circle_along_edges-29] In a polytope with a tetrahedral vertex figure, a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path Clifford polygon is the chord set of a true geodesic great circle,^[r] circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an isocline.

[Clifford_polygon-31] The edge-path of an isocline may be called the 4-polytope's Clifford polygon, as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic Clifford displacement.

[{30/11}-gram-33] 
The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.
The 600-cell Petrie polygon is a helical ring which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a triacontagram {30/11} of 30 #11 chords linking pairs of vertices that are 11 vertices apart on a geodesic.^[14] The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).

[Petrie_polygons_of_the_120-cell-34] The regular skew 30-gon is the Petrie polygon of the 600-cell and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell Boerdijk–Coxeter helix rings (the Petrie polygons of the 600-cell):^[s] connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell).

[fractional_square_roots-36] The fractional square root chord lengths are given as decimal fractions where:
𝚽 ≈ 0.618 is the inverse golden ratio 1/φ
𝚫 = 1 - 𝚽 = 𝚽² ≈ 0.382
𝜀 = √𝚫²/2 ≈ 0.073
and the 120-cell edge-length 1/φ²√2 is:
𝛇 = √𝜀 ≈ 0.270
For example:
𝛇 = √0.𝜀 = √0.073~ ≈ 0.270

[dodecahedral_cell_metrics-37] In the dodecahedral cell of the unit-radius 120-cell, the length of the edge (the #1 chord of the 120-cell) is 1/φ²√2 ≈ 0.270. The orange vertices lie at the Cartesian coordinates (±φ³√8, ±φ³√8, ±φ³√8) relative to origin at the cell center. They form a cube (dashed lines) of edge length 1/φ√2 ≈ 0.437 (the pentagon diagonal, and the #2 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 #3 chord of the 120-cell). The diameter of the dodecahedron is √3/φ√2 ≈ 0.757 (the cube diagonal, and the #4 chord of the 120-cell).

[38] The face pentagon edge (the 120-cell edge, the #1 chord) is in the golden ratio φ ≈ 1.618 to the face pentagon diagonal (the #2 chord).^[v]

[great_dodecagon-39] The 120-cell has a great circle polygon of 6 edges (#1 chords) alternating with 6 dodecahedron cell-diameters (#4 chords).^[v] This great circle polygon is an irregular dodecagon {12} with alternating edges of two different sizes. Two great hexagons with edges of a third size (√1, the #5 chord) are inscribed in the dodecagon.^[y] The {12} great circle is contained within a helical zig-zag of {30} edges (#1 chords) which is the Petrie polygon of the 120-cell.^[k] Hexagon vertices are 5 zig-zag edges apart. The 6 hexagon edges (#5 chords) and the 6 120-cell edges (#1 chords) of the dodecagon lie on the great circle, but the other 24 zig-zag edges (#1 chords) bridging the 6 cell diameters (#4 chords) do not lie on this great circle.

[great_hexagon-40] 
Triacontagram {30/5}=5{6}, the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.
Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a 0-gon great circle axis that does not intersect any vertices.^[k] There are 5 great hexagons inscribed in each Petrie polygon, in 5 different central planes.

[orthogonal_Petrie_polygons.-41] The Petrie polygon of the 5-cell is the pentagon (5-gon), and the Petrie polygon of the 120-cell is the triacontagon (30-gon).^[p] Each 120-cell Petrie 30-gon lies completely orthogonal to six 5-cell Petrie 5-gons, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.^[d]

[43] 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.^[16]

[46] Dodecahedra emerge as visible features in the 120-cell, but they also occur in the 600-cell as interior polytopes.^[18]

[50] In the 120-cell, each 24-cell belongs to two different 600-cells.^[20] The 120-cell can be partitioned into 25 disjoint 24-cells.^[21]

[56] 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.

[truncated_apex-57] Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular 5-cell with a regular tetrahedron base.

[58] 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.

[72] The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the diminishment of the 120-point 600-cell by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point snub 24-cell. Similarly, the 8-cell tesseract can be seen as a 16-point diminished 24-cell from which one 8-point 16-cell has been removed.

[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

[FOOTNOTECoxeter1973292–293Table_I(ii);_"120-cell"-3] Coxeter 1973, pp. 292–293, Table I(ii); "120-cell".

[FOOTNOTEDechant202118''Remark_5.7''-8] Dechant 2021, p. 18, Remark 5.7, explains why not.

[FOOTNOTEDechant2021Abstract-9] Dechant 2021, Abstract; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."

[10] Mathematics and Its History, John Stillwell, 1989, 3rd edition 2010, ISBN 0-387-95336-1

[FOOTNOTEStillwell2001-11] Stillwell 2001.

[FOOTNOTECoxeter1973156–157§8.7_Cartesian_coordinates-12] Coxeter 1973, pp. 156–157, §8.7 Cartesian coordinates.

[FOOTNOTEMamonePileioLevitt20101442Table_3-13] Mamone, Pileio & Levitt 2010, p. 1442, Table 3.

[FOOTNOTEMamonePileioLevitt20101433§4.1-14] Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors $\left(w,x,y,z\right)_{1}$ and $\left(w,x,y,z\right)_{2}$ according to
${\begin{pmatrix}w_{2}\\x_{2}\\y_{2}\\z_{2}\end{pmatrix}}*{\begin{pmatrix}w_{1}\\x_{1}\\y_{1}\\z_{1}\end{pmatrix}}={\begin{pmatrix}{w_{2}w_{1}-x_{2}x_{1}-y_{2}y_{1}-z_{2}z_{1}}\\{w_{2}x_{1}+x_{2}w_{1}+y_{2}z_{1}-z_{2}y_{1}}\\{w_{2}y_{1}-x_{2}z_{1}+y_{2}w_{1}+z_{2}x_{1}}\\{w_{2}z_{1}+x_{2}y_{1}-y_{2}x_{1}+z_{2}w_{1}}\end{pmatrix}}$

[FOOTNOTEMamonePileioLevitt20101438–1439§4.5_Regular_Convex_4-Polytopes,_Table_2,_Symmetry_operations-18] Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the pentadecagram isoclinic rotation of an individual 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the same pentadecagram isoclinic rotation of the 120-cell, its characteristic rotation. Mamone's q3 is the #8 chord of the 120-cell, the 5-cell's √5/2 edge; q13 is the #1 chord of the 120-cell, its edge.

[FOOTNOTECoxeter1973304Table_VI_(iv):_𝐈𝐈_=_{5,3,3}-25] Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.

[FOOTNOTEMamonePileioLevitt20101438–1439§4.5_Regular_Convex_4-Polytopes,_Table_2,_Symmetry_group_𝛨<sub>4</sub>-30] Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄; the 120-cell has 7200 distinct simple rotations (and 7200 reflections).

[FOOTNOTESadoc2001577–578§2.5_The_30/11_symmetry:_an_example_of_other_kind_of_symmetries-32] Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries.

[FOOTNOTECoxeter1973147§8.1_The_simple_truncations_of_the_general_regular_polytope-35] Coxeter 1973, p. 147, §8.1 The simple truncations of the general regular polytope; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."

[FOOTNOTECopher20196§3.2_Theorem_3.4-42] Copher 2019, p. 6, §3.2 Theorem 3.4.

[FOOTNOTESchleimerSegerman2013-44] Schleimer & Segerman 2013.

[FOOTNOTECoxeter1973298Table_V:_(iii)_Sections_of_{3,3,5}_beginning_with_a_vertex-45] Coxeter 1973, p. 298, Table V: (iii) Sections of {3,3,5} beginning with a vertex.

[47] Carlo H. Séquin. "Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes". Retrieved March 13, 2023.

[FOOTNOTEvan_Ittersum2020435§4.3.5_The_two_600-cells_circumscribing_a_24-cell-48] van Ittersum 2020, p. 435, §4.3.5 The two 600-cells circumscribing a 24-cell.

[FOOTNOTEDenneyHookerJohnsonRobinson20205§2_The_Labeling_of_H4-49] Denney et al. 2020, p. 5, §2 The Labeling of H4.

[FOOTNOTECoxeter1973305Table_VII:_Regular_Compounds_in_Four_Dimensions-51] Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.

[FOOTNOTEWaegellAravind20143–4§2_Geometry_of_the_120-cell:_rays_and_bases-52] Waegell & Aravind 2014, pp. 3–4, §2 Geometry of the 120-cell: rays and bases.

[FOOTNOTEWaegellAravind20145–6-53] Waegell & Aravind 2014, pp. 5–6.

[FOOTNOTESullivan19914–5The_Dodecahedron-54] Sullivan 1991, pp. 4–5, The Dodecahedron.

[FOOTNOTECoxeterdu_ValFlatherPetrie19384-55] 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."

[FOOTNOTEKocaAl-AjmiOzdes_Koca2011986–9886._Dual_of_the_snub_24-cell-59] Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell.

[FOOTNOTECoxeter1973§1.8_Configurations-60] Coxeter 1973, §1.8 Configurations.

[FOOTNOTECoxeter1991117-61] Coxeter 1991, p. 117.

[FOOTNOTESullivan199115Other_Properties_of_the_120-cell-62] Sullivan 1991, p. 15, Other Properties of the 120-cell.

[FOOTNOTESchleimerSegerman201316§6.1._Layers_of_dodecahedra-63] Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.

[FOOTNOTECoxeter197019–23§9._The_120-cell_and_the_600-cell-64] Coxeter 1970, pp. 19–23, §9. The 120-cell and the 600-cell.

[FOOTNOTESchleimerSegerman201316–18§6.2._Rings_of_dodecahedra-65] Schleimer & Segerman 2013, pp. 16–18, §6.2. Rings of dodecahedra.

[FOOTNOTEBanchoff2013-66] Banchoff 2013.

[FOOTNOTEZamboj20216–12§2_Mathematical_background-67] Zamboj 2021, pp. 6–12, §2 Mathematical background.

[FOOTNOTEZamboj202123–29§5_Hopf_tori_corresponding_to_circles_on_B<sup>2</sup>-68] Zamboj 2021, pp. 23–29, §5 Hopf tori corresponding to circles on B².

[FOOTNOTEChilton1964-69] Chilton 1964.

[FOOTNOTEDechant202118–206._The_Coxeter_Plane-70] Dechant 2021, pp. 18–20, 6. The Coxeter Plane.

[FOOTNOTEDenneyHookerJohnsonRobinson2020-71] Denney et al. 2020.

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