120cell  

Type  Convex regular 4polytope 
Schläfli symbol  {5,3,3} 
Coxeter diagram  
Cells  120 {5,3} 
Faces  720 {5} 
Edges  1200 
Vertices  600 
Vertex figure  tetrahedron 
Petrie polygon  30gon 
Coxeter group  H_{4}, [3,3,5] 
Dual  600cell 
Properties  convex, isogonal, isotoxal, isohedral 
Uniform index  32 
In geometry, the 120cell is the convex regular 4polytope (fourdimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C_{120}, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron^{[1]} and hecatonicosahedroid.^{[2]}
The boundary of the 120cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. Together they form 720 pentagonal faces, 1200 edges, and 600 vertices. It is the 4dimensional analogue of the regular dodecahedron, since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the dodecaplex has 120 dodecahedral facets, with 3 around each edge.^{[a]} Its dual polytope is the 600cell.
The 120cell 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 4polytope,^{[c]} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5cell,^{[d]} which is not found in any of the others.^{[4]} The 120cell is a fourdimensional Swiss Army knife: it contains one of everything.
It is daunting but instructive to study the 120cell, because it contains examples of every relationship among all the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.^{[5]} That is why Stillwell titled his paper on the 4polytopes and the history of mathematics^{[6]} of more than 3 dimensions The Story of the 120cell.^{[7]}
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  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 30gons x 6  20 30gons x 4  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
Natural Cartesian coordinates for a 4polytope centered at the origin of 4space occur in different frames of reference, depending on the long radius (centertovertex) chosen.
The 120cell 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})  24cell  600point 120cell 

64  ({±φ, ±φ, ±φ, ±φ^{−2}})  
64  ({±1, ±1, ±1, ±√5})  
64  ({±φ^{−1}, ±φ^{−1}, ±φ^{−1}, ±φ^{2}})  
96  ([0, ±φ^{−1}, ±φ, ±√5])  Snub 24cell  
96  ([0, ±φ^{−2}, ±1, ±φ^{2}])  Snub 24cell  
192  ([±φ^{−1}, ±1, ±φ, ±2]) 
where φ (also called 𝝉)^{[f]} is the golden ratio, 1 + √5/2 ≈ 1.618.
The unitradius 120cell has edge length 1/φ^{2}√2 ≈ 0.270.
In this frame of reference the 120cell lies vertex up in standard orientation, and its coordinates^{[9]} are the {permutations} and [even permutations] in the left column below:
120  8  ({±1, 0, 0, 0})  16cell  24cell  600cell  120cell 

16  ({±1, ±1, ±1, ±1}) / 2  Tesseract  
96  ([0, ±φ^{−1}, ±1, ±φ]) / 2  Snub 24cell  
480  Diminished 120cell  5point 5cell  24cell  600cell  
32  ([±φ, ±φ, ±φ, ±φ^{−2}]) / √8  (1, 0, 0, 0) (−1, √5, √5, √5) / 4 
({±√1/2, ±√1/2, 0, 0})  ({±1, 0, 0, 0}) ({±1, ±1, ±1, ±1}) / 2  
32  ([±1, ±1, ±1, ±√5]) / √8  
32  ([±φ^{−1}, ±φ^{−1}, ±φ^{−1}, ±φ^{2}]) / √8  
96  ([0, ±φ^{−1}, ±φ, ±√5]) / √8  
96  ([0, ±φ^{−2}, ±1, ±φ^{2}]) / √8  
192  ([±φ^{−1}, ±1, ±φ, ±2]) / √8  
The unitradius coordinates of uniform convex 4polytopes are related by quaternion multiplication. Since the regular 4polytopes are compounds of each other, their sets of Cartesian 4coordinates (quaternions) are set products of each other. The unitradius coordinates of the 600 vertices of the 120cell (in the left column above) are all the possible quaternion products^{[10]} of the 5 vertices of the 5cell, the 24 vertices of the 24cell, and the 120 vertices of the 600cell (in the other three columns above).^{[g]} 
The table gives the coordinates of at least one instance of each 4polytope, but the 120cell contains multiplesoffive inscribed instances of each of its precursor 4polytopes, occupying different subsets of its vertices. The (600point) 120cell is the convex hull of 5 disjoint (120point) 600cells. Each (120point) 600cell is the convex hull of 5 disjoint (24point) 24cells, so the 120cell is the convex hull of 25 disjoint 24cells. Each 24cell is the convex hull of 3 disjoint (8point) 16cells, so the 120cell is the convex hull of 75 disjoint 16cells. Uniquely, the (600point) 120cell is the convex hull of 120 disjoint (5point) 5cells.^{[k]}
See also: 600cell § Golden chords 
The 600point 120cell has all 8 of the 120point 600cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5cells.^{[d]} These two additional chords give the 120cell its characteristic isoclinic rotation,^{[ab]} in addition to all the rotations of the other regular 4polytopes which it inherits.^{[14]} They also give the 120cell a characteristic great circle polygon: an irregular great hexagon in which three 120cell edges alternate with three 5cell edges.^{[p]}
The 120cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600cell, 24cell, and 16cell do. Like the edges of the 5cell and the 8cell tesseract, they form zigzag Petrie polygons instead.^{[aa]} The 120cell's Petrie polygon is a triacontagon {30} zigzag skew polygon.^{[ac]}
Since the 120cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter.^{[ai]} Every regular convex 4polytope is inscribed in the 120cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4polytopes 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 4polytopes, two irregular 4polytopes occur naturally in the sequence of nested 4polytopes: the 96point snub 24cell and the 480point diminished 120cell.^{[c]}
The second thing to notice is that each numbered row (each chord) is marked with a triangle △, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ characteristic of the 16cell, great hexagons and great triangles △ characteristic of the 24cell, great decagons and great pentagons 𝜙 characteristic of the 600cell, and skew pentagrams ✩ or decagrams characteristic of the 5cell which are Petrie polygons that circle through a set of central planes and form face polygons but not great polygons.^{[s]}
Chords of the 120cell and its inscribed 4polytopes^{[15]}  

Inscribed^{[am]}  5cell  16cell  8cell  24cell  Snub  600cell  Dimin  120cell  
Vertices  5  8  16  24  96  120  480  600^{[j]}  
Edges  10^{[p]}  24  32  96  432  720  1200  1200^{[p]}  
Edge chord  #8^{[d]}  #7  #5  #5  #3  #3^{[t]}  #1  #1^{[ac]}  
Isocline chord^{[n]}  #8  #15  #10  #10  #5  #5  #4  #4^{[y]}  
Clifford polygon^{[ah]}  {5/2}  {8/3}  {6/2}  {15/2}  {15/4}^{[ab]}  
Chord  Arc  Edge  
#1 △ 
30  120cell edge^{[ac]}  1 1200^{[ab]} 
4 {3,3}  
15.5~°  √0.𝜀^{[ao]}  0.270~  
#2 ☐ 
15  face diagonal^{[ar]}  3600 
12 2{3,4}  
25.2~°  √0.19~  0.437~  
#3 𝜙 
10  𝝅/5  great decagon  10^{[k]} 720 
7200 
24 2{3,5}  
36°  √0.𝚫  0.618~  
#4 △ 
15/2  ^{[q]}  cell diameter^{[ap]}  1200 
4 {3,3}  
44.5~°  √0.57~  0.757~  
#5 △ 
6  𝝅/3  great hexagon^{[at]}  32 
225^{[k]} 96 
225 
5^{[k]} 1200 
2400^{[as]} 
32 4{4,3}  
60°  √1  1  
#6 𝜙 
5  2𝝅/5  great pentagon^{[v]}  720 
7200 
24 2{3,5}  
72°  √1.𝚫  1.175~  
#7 ☐ 
30/7  𝝅/2  great square^{[j]}  675^{[j]} 24 
675 48 
72 
1800 
16200 
54 9{3,4}  
90°  √2  1.414~  
#8 ✩ 
15/4  5cell^{[au]}  120^{[d]} 10 
720 
1200^{[ab]} 
4 {3,3}  
104.5~°  √2.5  1.581~  
#9 𝜙 
10/3  3𝝅/5  golden section  720 
7200 
24 2{3,5}  
108°  √2.𝚽  1.618~  
#10 △ 
3  2𝝅/3  great triangle  32 
25^{[k]} 96 
1200 
2400 
32 4{4,3}  
120°  √3  1.732~  
#11 ✩ 
30/11  {30/11}gram^{[an]}  1200 
4 {3,3}  
135.5~°  √3.43~  1.851~  
#12 𝜙 
5/2  4𝝅/5  great pent diag^{[au]}  720 
7200 
24 2{3,5}  
144°^{[a]}  √3.𝚽  1.902~  
#13 ✩ 
30/13  {30/13}gram  3600 
12 2{3,4}  
154.8~°  √3.81~  1.952~  
#14 △ 
15/7  {30/14}=2{15/7}  1200 
4 {3,3}  
164.5~°  √3.93~  1.982~  
#15 △☐𝜙 
2  𝝅  diameter  75^{[k]} 4 
8 
12 
48 
60 
240 
300^{[j]} 
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 120cell. All of the 2polytopes, 3polytopes and 4polytopes in the 120cell are made from the 15 1polytopes in the table.
The black integers in table cells are incidence counts of the row's chord in the column's 4polytope. For example, in the #3 chord row, the 600cell's 72 great decagons contain 720 #3 chords in all.
The red integers are the number of disjoint 4polytopes above (the column label) which compounded form a 120cell. For example, the 120cell is a compound of 25 disjoint 24cells (25 * 24 vertices = 600 vertices).
The green integers are the number of distinct 4polytopes above (the column label) which can be picked out in the 120cell. For example, the 120cell contains 225 distinct 24cells which share components.
The blue integers in the right column are incidence counts of the row's chord at each 120cell vertex. For example, in the #3 chord row, 24 #3 chords converge at each of the 120cell'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 120cell.
The 120cell is the compound of all five of the other regular convex 4polytopes.^{[19]} All the relationships among the regular 1, 2, 3 and 4polytopes occur in the 120cell.^{[b]} It is a fourdimensional jigsaw puzzle in which all those polytopes are the parts.^{[20]} Although there are many sequences in which to construct the 120cell by putting those parts together, ultimately they only fit together one way. The 120cell is the unique solution to the combination of all these polytopes.^{[7]}
The regular 1polytope occurs in only 15 distinct lengths in any of the component polytopes of the 120cell.^{[al]} By Alexandrov's uniqueness theorem, convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, so each regular 4polytope has its own unique subset of these 15 chords.
Only 4 of those 15 chords occur in the 16cell, 8cell and 24cell. The four hypercubic chords √1, √2, √3 and √4 are sufficient to build the 24cell and all its component parts. The 24cell 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 600cell. The four golden chords are square roots of irrational fractions that are functions of √5. The 600cell 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 600cell which do not occur in the 24cell are pentagons, and icosahedra.
See also: 600cell § Icosahedra 
All 15 chords, and 15 other distinct chordal distances enumerated below, occur in the 120cell. Notable among the new parts found in the 120cell which do not occur in the 600cell are regular 5cells and √5/2 chords.^{[aw]} The relationships between the regular 5cell (the simplex regular 4polytope) and the other regular 4polytopes are manifest directly only in the 120cell.^{[i]} The 600point 120cell is a compound of 120 disjoint 5point 5cells, and it is also a compound of 5 disjoint 120point 600cells (two different ways). Each 5cell has one vertex in each of 5 disjoint 600cells, and therefore in each of 5 disjoint 24cells, 5 disjoint 8cells, and 5 disjoint 16cells.^{[ba]} Each 5cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4polytopes.^{[w]}
See also: 5cell § Geodesics and rotations 
The 30 distinct chords^{[al]} found in the 120cell 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,^{[q]} 𝜙 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 120cell, beginning with a vertex, the 0_{0} section. The correspondence is that each 120cell 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.^{[bb]} The #1 chord is the "radius" of the 1_{0} section, the tetrahedral vertex figure of the 120cell.^{[ar]} The #14 chord is the "radius" of its congruent opposing 29_{0} section. The #7 chord is the "radius" of the central section of the 120cell, in which two opposing 15_{0} sections are coincident.
30 chords (15 180° pairs) make 15 kinds of great circle polygons and polyhedral sections^{[22]}  

Short chord  Great circle polygons  Rotation  Long chord  
1_{0} #1 
^{[af]}  400 irregular great hexagons^{[q]} / 4 (600 great rectangles) 
4𝝅^{[l]} {15/4}^{[ab]} #4^{[y]} 
29_{0} #14  
15.5~°  √0.𝜀^{[ao]}  0.270~  164.5~°  √3.93~  1.982~  
2_{0} #2 
^{[ar]}  Great rectangles in ☐ planes 
4𝝅 {30/13} #13 
28_{0} #13  
25.2~°  √0.19~  0.437~  154.8~°  √3.81~  1.952~  
3_{0} #3 
720 great decagons / 12 (3600 great rectangles) in 720 𝜙 planes 
5𝝅 {15/2} #5 
27_{0} #12  
36°  √0.𝚫  0.618~  144°^{[a]}  √3.𝚽  1.902~  
4_{0} #4−1 
Great rectangles in ☐ planes 
26_{0} #11+1  
41.4~°  √0.5  0.707~  138.6~°  √3.5  1.871~  
5_{0} #4 
200 irregular great dodecagons^{[be]} / 4 (600 great rectangles) in 200 △ planes 
^{[bd]}  25_{0} #11  
44.5~°  √0.57~  0.757~  135.5~°  √3.43~  1.851~  
6_{0} #4+1 
Great rectangles in ☐ planes 
24_{0} #11−1  
49.1~°  √0.69~  0.831~  130.9~°  √3.31~  1.819~  
7_{0} #5−1 
Great rectangles in ☐ planes 
23_{0} #10+1  
56°  √0.88~  0.939~  124°  √3.12~  1.766~  
8_{0} #5 
400 regular great hexagons^{[at]} / 16 (1200 great rectangles) in 200 △ planes 
4𝝅^{[l]} 2{10/3} #4 
22_{0} #10  
60°  √1  1  120°  √3  1.732~  
9_{0} #5+1 
Great rectangles in ☐ planes 
21_{0} #10−1  
66.1~°  √1.19~  1.091~  113.9~°  √2.81~  1.676~  
10_{0} #6−1 
Great rectangles in ☐ planes 
20_{0} #9+1  
69.8~°  √1.31~  1.144~  110.2~°  √2.69~  1.640~  
11_{0} #6 
1440 great pentagons^{[v]} / 12 (3600 great rectangles) in 720 𝜙 planes 
4𝝅 {24/5} #9 
19_{0} #9  
72°  √1.𝚫  1.175~  108°  √2.𝚽  1.618~  
12_{0} #6+1 
1200 great digon 5cell edges^{[bf]} / 4 (600 great rectangles) in 200 △ planes 
4𝝅^{[l]} {5/2} #8 
18_{0} #8  
75.5~°  √1.5  1.224~  104.5~°  √2.5  1.581~  
13_{0} #6+2 
Great rectangles in ☐ planes 
17_{0} #8−1  
81.1~°  √1.69~  1.300~  98.9~°  √2.31~  1.520~  
14_{0} #7−1 
Great rectangles in ☐ planes 
16_{0} #7+1  
84.5~°  √0.81~  1.345~  95.5~°  √2.19~  1.480~  
15_{0} #7 
4050 great squares^{[j]} / 27 in 4050 ☐ planes 
4𝝅 {30/7} #7 
15_{0} #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^{[n]} of a distinct class,^{[r]} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.^{[bl]} 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.^{[bm]} In each class of rotation,^{[bk]} vertices rotate on a distinct kind of circular geodesic isocline^{[m]} which has a characteristic circumference, skew Clifford polygram^{[ah]} and chord number, listed in the Rotation column above.^{[ag]}
Considering the adjacency matrix of the vertices representing the polyhedral graph of the unitradius 120cell, the graph diameter is 15, connecting each vertex to its coordinatenegation 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/φ^{2}√2 ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582.
The vertices of the 120cell polyhedral graph are 3colorable.
The graph is Eulerian having degree 4 in every vertex. Its edge set can be decomposed into two Hamiltonian cycles.^{[25]}
The 120cell is the sixth in the sequence of 6 convex regular 4polytopes (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 600cell,^{[h]} just as the 600cell can be deconstructed into twentyfive distinct instances (or five disjoint instances) of its predecessor the 24cell,^{[bn]} the 24cell can be deconstructed into three distinct instances of its predecessor the tesseract (8cell), and the 8cell can be deconstructed into two disjoint instances of its predecessor (and dual) the 16cell.^{[28]} The 120cell contains 675 distinct instances (75 disjoint instances) of the 16cell.^{[j]}
The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600cell's edge length is ~0.618 times its radius (the inverse golden ratio), but the 120cell's edge length is ~0.270 times its radius.
Since the 120cell is the dual of the 600cell, it can be constructed from the 600cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600cell of unit long radius, this results in a 120cell of slightly smaller long radius (φ^{2}/√8 ≈ 0.926) and edge length of exactly 1/4. Thus the unit edgelength 120cell (with long radius φ^{2}√2 ≈ 3.702) can be constructed in this manner just inside a 600cell of long radius 4. The unit radius 120cell (with edgelength 1/φ^{2}√2 ≈ 0.270) can be constructed in this manner just inside a 600cell of long radius √8/φ^{2} ≈ 1.080.
Reciprocally, the unitradius 120cell can be constructed just outside a 600cell of slightly smaller long radius φ^{2}/√8 ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600cell vertices. The 120cell whose coordinates are given above of long radius √8 = 2√2 ≈ 2.828 and edgelength 2/φ^{2} = 3−√5 ≈ 0.764 can be constructed in this manner just outside a 600cell of long radius φ^{2}, which is smaller than √8 in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600cell, so that must be φ. The 120cell of edgelength 2 and long radius φ^{2}√8 ≈ 7.405 given by Coxeter^{[3]} can be constructed in this manner just outside a 600cell of long radius φ^{4} and edgelength φ^{3}.
Therefore, the unitradius 120cell can be constructed from its predecessor the unitradius 600cell in three reciprocation steps.
Since the 120cell contains inscribed 600cells, it contains its own dual of the same radius. The 120cell contains five disjoint 600cells (ten overlapping inscribed 600cells of which we can pick out five disjoint 600cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600cell are vertices of the 120cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600cells.
The dodecahedral cells of the 120cell have tetrahedral cells of the 600cells inscribed in them.^{[30]} Just as the 120cell is a compound of five 600cells (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).^{[31]} This shows that the 120cell contains, among its many interior features, 120 compounds of ten tetrahedra, each of which is dimensionally analogous to the whole 120cell as a compound of ten 600cells.^{[h]}
All ten tetrahedra can be generated by two chiral fiveclick rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600cells inscribed in the 120cell.^{[bo]} Therefore the whole 120cell, with all ten inscribed 600cells, can be generated from just one 600cell by rotating its cells.
Another consequence of the 120cell containing inscribed 600cells is that it is possible to construct it by placing 4pyramids of some kind on the cells of the 600cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron.^{[bp]}
Only 120 tetrahedral cells of each 600cell can be inscribed in the 120cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedroninscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others facebonded around it lying only partially within the dodecahedron. The central tetrahedron is edgebonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.^{[bq]} The central cell is vertexbonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.
Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits of order 120.^{[33]} The following describe and 24cells 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
With quaternions where is the conjugate of and and , then the Coxeter group is the symmetry group of the 600cell and the 120cell of order 14400.
Given such that and as an exchange of within , we can construct:
This configuration matrix represents the 120cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[34]}^{[35]}
Here is the configuration expanded with kface elements and kfigures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.
H_{4}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  kfig  Notes  

A_{3}  ( )  f_{0}  600  4  6  4  {3,3}  H_{4}/A_{3} = 14400/24 = 600  
A_{1}A_{2}  { }  f_{1}  2  1200  3  3  {3}  H_{4}/A_{2}A_{1} = 14400/6/2 = 1200  
H_{2}A_{1}  {5}  f_{2}  5  5  720  2  { }  H_{4}/H_{2}A_{1} = 14400/10/2 = 720  
H_{3}  {5,3}  f_{3}  20  30  12  120  ( )  H_{4}/H_{3} = 14400/120 = 120 
The 120cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings (discrete Hopf fibration).^{[36]}
The cell locations lend themselves to a hyperspherical description.^{[37]} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12).
Starting at the North Pole, we can build up the 120cell in 9 latitudinal layers, with allusions to terrestrial 2sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2sphere, with the equatorial centroids lying on a great 2sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles.
Layer #  Number of Cells  Description  Colatitude  Region 

1  1 cell  North Pole  0°  Northern Hemisphere 
2  12 cells  First layer of meridional cells / "Arctic Circle"  36°  
3  20 cells  Nonmeridian / interstitial  60°  
4  12 cells  Second layer of meridional cells / "Tropic of Cancer"  72°  
5  30 cells  Nonmeridian / interstitial  90°  Equator 
6  12 cells  Third layer of meridional cells / "Tropic of Capricorn"  108°  Southern Hemisphere 
7  20 cells  Nonmeridian / interstitial  120°  
8  12 cells  Fourth layer of meridional cells / "Antarctic Circle"  144°  
9  1 cell  South Pole  180°  
Total  120 cells 
The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell.
The 120cell can be partitioned into 12 disjoint 10cell great circle rings, forming a discrete/quantized Hopf fibration.^{[38]}^{[39]}^{[40]}^{[41]}^{[36]} Starting with one 10cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10cell rings can be placed adjacent to the original 10cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical torsion. They are all equivalent. The spiraling is a result of the 3sphere curvature. The inner ring and the five outer rings now form a six ring, 60cell solid torus. One can continue adding 10cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120cell, like the 3sphere, is the union of these two (Clifford) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.^{[42]} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint (Clifford parallel) great circles.
There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter chords, forming an irregular dodecagon in a central plane.^{[q]} Both these great circle paths have dual great circle paths in the 600cell. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600cell, forming a decagon.^{[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 600cell. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub 24cell (or icosahedral pyramids in the 600cell), forming a hexagon.
Another great circle polygon path exists which is unique to the 120cell and has no dual counterpart in the 600cell. This path consists of 3 120cell edges alternating with 3 inscribed 5cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges illustrated above.^{[p]} Each 5cell edge runs through the volume of three dodecahedral cells (in a ring of ten facebonded 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.^{[q]}
Projections to 3D of a 4D 120cell performing a simple rotation  

From outside the 3sphere in 4space.  Inside the 3D surface of the 3sphere. 
As in all the illustrations in this article, only the edges of the 120cell appear in these renderings. All the other chords are not shown. The complex interior parts of the 120cell, all its inscribed 600cells, 24cells, 8cells, 16cells and 5cells, 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 120cell 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 3sphere. Both these methods distort the object, because the cells are not actually nested inside each other (they meet facetoface), 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).
Projection  Dodecahedron  120cell 

Schlegel diagram  12 pentagon faces in the plane 
120 dodecahedral cells in 3space 
Stereographic projection  With transparent faces 
Orthogonal projections of the 120cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30gonal projection was made in 1963 by B. L. Chilton.^{[44]}
The H3 decagonal projection shows the plane of the van Oss polygon.
H_{4}    F_{4} 

[30] (Red=1) 
[20] (Red=1) 
[12] (Red=1) 
H_{3}  A_{2} / B_{3} / D_{4}  A_{3} / B_{2} 
[10] (Red=5, orange=10) 
[6] (Red=1, orange=3, yellow=6, lime=9, green=12) 
[4] (Red=1, orange=2, yellow=4, lime=6, green=8) 
3dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image.
3D isometric projection 
Animated 4D rotation 
The 120cell is one of 15 regular and uniform polytopes with the same H_{4} symmetry [3,3,5]:^{[46]}
H_{4} family polytopes  

120cell  rectified 120cell 
truncated 120cell 
cantellated 120cell 
runcinated 120cell 
cantitruncated 120cell 
runcitruncated 120cell 
omnitruncated 120cell  
{5,3,3}  r{5,3,3}  t{5,3,3}  rr{5,3,3}  t_{0,3}{5,3,3}  tr{5,3,3}  t_{0,1,3}{5,3,3}  t_{0,1,2,3}{5,3,3}  
600cell  rectified 600cell 
truncated 600cell 
cantellated 600cell 
bitruncated 600cell 
cantitruncated 600cell 
runcitruncated 600cell 
omnitruncated 600cell  
{3,3,5}  r{3,3,5}  t{3,3,5}  rr{3,3,5}  2t{3,3,5}  tr{3,3,5}  t_{0,1,3}{3,3,5}  t_{0,1,2,3}{3,3,5} 
The 120cell is similar to three regular 4polytopes: the 5cell {3,3,3} and tesseract {4,3,3} of Euclidean 4space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure {3,3}:
{p,3,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Paracompact  Noncompact  
Name  {3,3,3}  {4,3,3}  {5,3,3}  {6,3,3}  {7,3,3}  {8,3,3}  ...{∞,3,3}  
Image  
Cells {p,3} 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 
The 120cell is a part of a sequence of 4polytopes and honeycombs with dodecahedral cells:
{5,3,p} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
Name  {5,3,3}  {5,3,4}  {5,3,5}  {5,3,6}  {5,3,7}  {5,3,8}  ... {5,3,∞} 
Image  
Vertex figure 
{3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞} 
Since the 600point 120cell has 5 disjoint inscribed 600cells, it can be diminished by the removal of one of those 120point 600cells, creating an irregular 480point 4polytope.^{[bt]}
Each dodecahedral cell of the 120cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16point 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 120cell is the tetrahedron,^{[bp]} 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 120cell and the tetrahedrally diminished 120cell both have 1200 edges.
The 480point diminished 120cell may be called the tetrahedrally diminished 120cell because its cells are tetrahedrally diminished, or the 600cell diminished 120cell because the vertices removed formed a 600cell inscribed in the 120cell, or even the regular 5cells diminished 120cell because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5cells, leaving 120 regular tetrahedra.^{[d]}
The Davis 120cell, introduced by Davis (1985), is a compact 4dimensional hyperbolic manifold obtained by identifying opposite faces of the 120cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4dimensional hyperbolic space.