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

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]

where φ (also called 𝝉)^[f] 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 in standard orientation, and its coordinates^[9] are the {permutations} and [even permutations] in the left column below:

The table gives the coordinates of at least one instance of each 4-polytope, 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,^[ab] in addition to all the rotations of the other regular 4-polytopes which it inherits.^[14] 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.^[q]

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.^[aa] The 120-cell's Petrie polygon is a triacontagon {30} zig-zag skew polygon.^[ac]

Since the 120-cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter.^[ai] Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.^[al]

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]

The second thing to notice is that each numbered row is marked with a triangle △, square ☐, or pentagon ✩. The 15 chords lie in central planes of three kinds: great square ☐ planes characteristic of the 16-cell, great hexagon and great triangle △ planes characteristic of the 24-cell, or great decagon and great pentagon ✩ planes characteristic of the 600-cell.^[k]

Chords of the 120-cell and its inscribed 4-polytopes[15]
Inscribed[am]				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[q]	24	32	96	432	720	1200	1200[q]
Edge chord				#8[d]	#7	#5	#5	#3	#3[t]	#1	#1[ac]
Isocline chord[o]				#8	#15	#10	#10	#5	#5	#4	#4[y]
Clifford polygon[ah]				{5/2}	{8/3}		{6/2}		{15/2}		{15/4}[ab]
#1 △	𝝅/0.270~	edge[ac]									1 1200[ab]	4 {3,3}
	15.5~°	√0.𝜀[ao]	0.270~
#2 ☐		face diagonal[ar]									3600	12 2{3,4}
	25.2~°	√0.19~	0.437~
#3 ✩	𝝅/5	great decagon edge ${\displaystyle \phi ^{-1))$							10[j] 720		7200	24 2{3,5}
	36°	√0.𝚫	0.618~
#4 △	[r]	cell diameter[ap]									1200	4 {3,3}
	44.5~°	√0.57~	0.757~
#5 △	𝝅/3	great hexagon[at] edge				32	225[j] 96	225	5[j] 1200		2400[as]	32 4{4,3}
	60°	√1	1
#6 ✩	2𝝅/5	great pentagon[v] edge							720		7200	24 2{3,5}
	72°	√1.𝚫	1.175~
#7 ☐	𝝅/2	great square[i] edge			675[i] 24	675 48	72		1800		16200	54 9{3,4}
	90°	√2	1.414~
#8 △	𝝅/1.823~	5-cell edge[au]		120[d] 10						720	1200[ab]	4 {3,3}
	104.5~°	√2.5	1.581~
#9 ✩	3𝝅/5	golden section ${\displaystyle \phi }$							720		7200	24 2{3,5}
	108°	√2.𝚽	1.618~
#10 △	2𝝅/3	great triangle edge				32	25[j] 96		1200		2400	32 4{4,3}
	120°	√3	1.732~
#11 △		{30/11}-gram[an] edge									1200	4 {3,3}
	135.5~°	√3.43~	1.851~
#12 ✩	4𝝅/5	great pentagon diagonal							720		7200	24 2{3,5}
	144°[a]	√3.𝚽	1.902~
#13 ☐		{30/13}-gram edge									3600	12 2{3,4}
	154.8~°	√3.81~	1.952~
#14 △		{30/14}=2{15/7} edge									1200	4 {3,3}
	164.5~°	√3.93~	1.982~
#15 △☐✩	𝝅	diameter			75[j] 4	8	12	48	60	240	300[i]	1
	180°	√4	2
Squared lengths total[av]				25	64	256	576		14400		360000[al]	300

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

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 120-cell vertex. For example, in the #3 chord row, 24 #3 chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral vertex figure 2{3,5}. In total 300 major chords^[al] of 15 distinct lengths meet at each vertex of the 120-cell.

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.^[19] 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 these polytopes.^[7]

The regular 1-polytope occurs in only 15 distinct lengths in any of the component polytopes of the 120-cell.^[al]

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 these 4 chords and all the regular polytopes 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 √5. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes 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, and 15 other distinct chordal distances enumerated below, occur in 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.^[aw] The relationships between the regular 5-cell (the simplex regular 4-polytope) and the other regular 4-polytopes are manifest only in the 120-cell.

Geodesic rectangles

The 30 distinct chords^[al] found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: △ planes that intersect {12} vertices in an irregular dodecagon,^[r] ✩ planes that intersect {10} vertices in a regular decagon, and ☐ planes that intersect {4} vertices in several kinds of rectangle, including a square.

Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges.

Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing polyhedral sections of the 120-cell, beginning with a vertex, the 0₀ section. The correspondence is that each 120-cell vertex is surrounded by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius.^[ax] The #1 chord is the "radius" of the 1₀ section, the tetrahedral vertex figure of the 120-cell.^[ar] The #14 chord is the "radius" of its congruent opposing 29₀ section. The #7 chord is the "radius" of the central section of the 120-cell, in which two opposing 15₀ sections are coincident.

30 chords (15 180° pairs) make 15 kinds of great circle polygons and polyhedral sections[21]
Short chord				Great circle polygons		Rotation	Long chord
10 #1	[ay]	${\displaystyle 1/\phi ^{2}{\sqrt {2))}$			400 irregular great hexagons[r] / 4 (600 great rectangles) in 200 △ planes	4𝝅[m] {15/4}[ab] #4[y]		${\displaystyle \phi ^{5}{\sqrt {3))/{\sqrt {8))}$		290 #14
	15.5~°	√0.𝜀[ao]	0.270~				164.5~°	√3.93~	1.982~
20 #2	[ar]	${\displaystyle 1/\phi {\sqrt {2))}$			Great rectangles in ☐ planes	8𝝅 {30/13} #13				280 #13
	25.2~°	√0.19~	0.437~				154.8~°	√3.81~	1.952~
30 #3	${\displaystyle \pi /5}$	${\displaystyle 1/\phi }$			720 great decagons / 12 (3600 great rectangles) in 720 ✩ planes	5𝝅 {15/2} #5	${\displaystyle 4\pi /5}$	${\displaystyle {\sqrt {2+\phi ))}$		270 #12
	36°	√0.𝚫	0.618~				144°[a]	√3.𝚽	1.902~
40 #4−1		${\displaystyle {\sqrt {1))/{\sqrt {2))}$			Great rectangles in ☐ planes			${\displaystyle {\sqrt {7))/{\sqrt {2))}$		260 #11+1
	41.4~°	√0.5	0.707~				138.6~°	√3.5	1.871~
50 #4		${\displaystyle {\sqrt {3))/\phi {\sqrt {2))}$			200 irregular great dodecagons[ba] / 4 (600 great rectangles) in 200 △ planes	[az]		${\displaystyle \phi ^{2}/{\sqrt {2))}$		250 #11
	44.5~°	√0.57~	0.757~				135.5~°	√3.43~	1.851~
60 #4+1					Great rectangles in ☐ planes					240 #11−1
	49.1~°	√0.69~	0.831~				130.9~°	√3.31~	1.819~
70 #5−1					Great rectangles in ☐ planes					230 #10+1
	56°	√0.88~	0.939~				124°	√3.12~	1.766~
80 #5	${\displaystyle \pi /3}$				400 regular great hexagons[at] / 16 (1200 great rectangles) in 200 △ planes	4𝝅[m] 2{10/3} #4	${\displaystyle 2\pi /3}$			220 #10
	60°	√1	1				120°	√3	1.732~
90 #5+1					Great rectangles in ☐ planes					210 #10−1
	66.1~°	√1.19~	1.091~				113.9~°	√2.81~	1.676~
100 #6−1					Great rectangles in ☐ planes					200 #9+1
	69.8~°	√1.31~	1.144~				110.2~°	√2.69~	1.640~
110 #6	${\displaystyle 2\pi /5}$	${\displaystyle {\sqrt {3-\phi ))}$			1440 great pentagons[v] / 12 (3600 great rectangles) in 720 ✩ planes	8𝝅 {24/5} #9	${\displaystyle 3\pi /5}$	${\displaystyle \phi }$		190 #9
	72°	√1.𝚫	1.175~				108°	√2.𝚽	1.618~
120 #6+1		${\displaystyle {\sqrt {3))/{\sqrt {2))}$			1200 great digon 5-cell edges[bb] / 4 (600 great rectangles) in 200 △ planes	4𝝅[m] {5/2} #8		${\displaystyle {\sqrt {5))/{\sqrt {2))}$		180 #8
	75.5~°	√1.5	1.224~				104.5~°	√2.5	1.581~
130 #6+2					Great rectangles in ☐ planes					170 #8−1
	81.1~°	√1.69~	1.300~				98.9~°	√2.31~	1.520~
140 #7−1					Great rectangles in ☐ planes					160 #7+1
	84.5~°	√0.81~	1.345~				95.5~°	√2.19~	1.480~
150 #7	${\displaystyle \pi /2}$				4050 great squares[i] / 27 in 4050 ☐ planes	8𝝅 {30/7} #7	${\displaystyle \pi /2}$			150 #7
	90°	√2	1.414~				90°	√2	1.414~

Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation^[o] of a distinct class,^[s] which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.^[bg] There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.^[bh] In each class of rotation,^[bf] vertices rotate on a distinct kind of circular geodesic isocline^[n] which has a characteristic circumference, skew Clifford polygram^[ah] and chord number, listed in the Rotation column above.^[ag]

Concentric hulls

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, 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.^[24]

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,^[h] just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the 24-cell,^[bi] 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 (and dual) the 16-cell.^[27] 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

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). 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.^[29] 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).^[30] 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.^[h]

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.^[bj] 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.^[bk]

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.^[bl] 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 ${\displaystyle O(\Lambda )=W(H_{4})=I}$ of order 120.^[32] 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 = ${\displaystyle T\oplus T'\oplus S'}$.

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.^[33][34]

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

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	1200	3	3	{3}	H4/A2A1 = 14400/6/2 = 1200
H2A1		{5}	f2	5	5	720	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.^[35]

Layered stereographic projection

The cell locations lend themselves to a hyperspherical description.^[36] 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.

Intertwining rings

The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.^{[37][38][39][40][35]} 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.^[41] 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, forming an irregular dodecagon in a central plane.^[r] Both these 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.^[t] The alternating cell/edge path 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).

Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges illustrated above.^[q] Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each irregular great dodecagon, in alternate positions.

Perspective projections

Projections to 3D of a 4D 120-cell performing a simple rotation
From outside the 3-sphere in 4-space.	Inside the 3D surface of the 3-sphere.

As in all the illustrations in this article, only the edges of the 120-cell appear in these renderings. All the other chords are not shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.

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.

A comparison of perspective projections of the 3D dodecahedron to 2D (below left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. Schlegel diagrams use perspective to show depth in the dimension which has been flattened, choosing a view point above a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. Stereographic projections use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a 3-sphere. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating animations above, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must).

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

Enhanced perspective projections
	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

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.^[43]

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

Orthographic projections by Coxeter planes^[44]

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

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]:^[45]

H4 family polytopes
120-cell	rectified 120-cell	truncated 120-cell	cantellated 120-cell	runcinated 120-cell	cantitruncated 120-cell	runcitruncated 120-cell	omnitruncated 120-cell
{5,3,3}	r{5,3,3}	t{5,3,3}	rr{5,3,3}	t0,3{5,3,3}	tr{5,3,3}	t0,1,3{5,3,3}	t0,1,2,3{5,3,3}
600-cell	rectified 600-cell	truncated 600-cell	cantellated 600-cell	bitruncated 600-cell	cantitruncated 600-cell	runcitruncated 600-cell	omnitruncated 600-cell
{3,3,5}	r{3,3,5}	t{3,3,5}	rr{3,3,5}	2t{3,3,5}	tr{3,3,5}	t0,1,3{3,3,5}	t0,1,2,3{3,3,5}

{p,3,3} polytopes

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

{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.^[bo]

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.^[aq] 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,^[bk] 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.