120cell
 REDIRECT Template:Infobox 4polytope
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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.^{[loweralpha 1]} Its dual polytope is the 600cell.
Geometry
The 120cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).^{[loweralpha 2]} As the sixth and largest regular convex 4polytope,^{[loweralpha 3]} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5cell,^{[loweralpha 4]} 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]}
Cartesian coordinates
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.
√8 radius coordinates
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)^{[loweralpha 6]} is the golden ratio, 1 + √5/2 ≈ 1.618.
Unit radius coordinates
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).^{[loweralpha 7]} 
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.^{[loweralpha 10]}
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.^{[loweralpha 4]} These two additional chords give the 120cell its characteristic isoclinic rotation,^{[loweralpha 23]} in addition to all the rotations of the other regular 4polytopes which it inherits.^{[13]} They also give the 120cell a characteristic great circle polygon: an irregular great hexagon in which three 120cell edges alternate with three 5cell edges.^{[loweralpha 17]}
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.^{[loweralpha 27]} The 120cell's Petrie polygon is a triacontagon {30} zigzag skew polygon.^{[loweralpha 30]}
Since the 120cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter.^{[loweralpha 28]} 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.^{[loweralpha 32]}
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.^{[loweralpha 3]}
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 16cell, great hexagon and great triangle △ planes characteristic of the 24cell, or great decagon and great pentagon ✩ planes characteristic of the 600cell.^{[loweralpha 11]}
Chords of the 120cell and its inscribed 4polytopes^{[14]}  

Inscribed^{[loweralpha 33]}  5cell  16cell  8cell  24cell  Snub  600cell  Dimin  120cell  
Vertices  5  8  16  24  96  120  480  600  
Edges  10^{[loweralpha 17]}  24  32  96  432  720  1200  1200^{[loweralpha 17]}  
Edge chord  #8^{[loweralpha 4]}  #7  #5  #5  #3  #3^{[loweralpha 20]}  #1  #1^{[loweralpha 30]}  
Isocline chord^{[loweralpha 15]}  #8  #15  #10  #10  #5  #5  #4  #4^{[loweralpha 35]}  
Clifford polygon^{[loweralpha 26]}  {5/2}  {8/3}  {6/2}  {15/2}  {15/4}^{[loweralpha 23]}  
#1 △ 
𝝅/0.270~  edge^{[loweralpha 30]}  1 1200^{[loweralpha 23]} 
4 {3,3}  
15.5~°  √0.𝜀^{[loweralpha 36]}  0.270~  
#2 ☐ 
face diagonal^{[loweralpha 39]}  3600 
12 2{3,4}  
25.2~°  √0.19~  0.437~  
#3 ✩ 
𝝅/5  great decagon edge [math]\displaystyle{ \phi^{1} }[/math]  10^{[loweralpha 10]} 720 
7200 
24 2{3,5}  
36°  √0.𝚫  0.618~  
#4 △ 
^{[loweralpha 18]}  cell diameter^{[loweralpha 37]}  1200 
4 {3,3}  
44.5~°  √0.57~  0.757~  
#5 △ 
𝝅/3  great hexagon^{[loweralpha 41]} edge  32 
225^{[loweralpha 10]} 96 
225 
5^{[loweralpha 10]} 1200 
2400^{[loweralpha 40]} 
32 4{4,3}  
60°  √1  1  
#6 ✩ 
2𝝅/5  great pentagon^{[loweralpha 21]} edge  720 
7200 
24 2{3,5}  
72°  √1.𝚫  1.175~  
#7 ☐ 
𝝅/2  great square^{[loweralpha 9]} edge  675^{[loweralpha 9]} 24 
675 48 
72 
1800 
16200 
54 9{3,4}  
90°  √2  1.414~  
#8 △ 
𝝅/1.823~  5cell edge^{[loweralpha 42]}  120^{[loweralpha 4]} 10 
720 
1200^{[loweralpha 23]} 
4 {3,3}  
104.5~°  √2.5  1.581~  
#9 ✩ 
3𝝅/5  golden section [math]\displaystyle{ \phi }[/math]  720 
7200 
24 2{3,5}  
108°  √2.𝚽  1.618~  
#10 △ 
2𝝅/3  great triangle edge  32 
25^{[loweralpha 10]} 96 
1200 
2400 
32 4{4,3}  
120°  √3  1.732~  
#11 △ 
{30/11}gram^{[loweralpha 34]} edge  1200 
4 {3,3}  
135.5~°  √3.43~  1.851~  
#12 ✩ 
4𝝅/5  great pentagon diagonal  720 
7200 
24 2{3,5}  
144°^{[loweralpha 1]}  √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^{[loweralpha 10]} 4 
8 
12 
48 
60 
240 
300^{[loweralpha 9]} 
1  
180°  √4  2  
Squared lengths total^{[loweralpha 43]}  25  64  256  576  14400  360000^{[loweralpha 32]}  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^{[loweralpha 32]} of 15 distinct lengths meet at each vertex of the 120cell.
Relationships among interior polytopes
The 120cell is the compound of all five of the other regular convex 4polytopes. All the relationships among the regular 1, 2, 3 and 4polytopes occur in the 120cell.^{[loweralpha 2]} It is a fourdimensional jigsaw puzzle in which all those polytopes are the parts.^{[17]} 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.^{[loweralpha 32]}
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.
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.^{[loweralpha 44]} The relationships between the regular 5cell (the simplex regular 4polytope) and the other regular 4polytopes are manifest only in the 120cell.
Geodesic rectangles
The 30 distinct chords^{[loweralpha 32]} 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: Template:Backgroundcolor in an irregular dodecagon,^{[loweralpha 18]} Template:Backgroundcolor in a regular decagon, and Template:Backgroundcolor 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.^{[loweralpha 45]} The #1 chord is the "radius" of the 1_{0} section, the tetrahedral vertex figure of the 120cell.^{[loweralpha 39]} 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^{[19]}  

Short chord  Great circle polygons  Rotation  Long chord  
1_{0} #1 
^{[loweralpha 46]}  [math]\displaystyle{ 1 / \phi^2\sqrt{2} }[/math]  400 irregular great hexagons^{[loweralpha 18]} / 4 (600 great rectangles) 
4𝝅^{[loweralpha 13]} {15/4}^{[loweralpha 23]} #4^{[loweralpha 35]} 
[math]\displaystyle{ \phi^{5}\sqrt{3} / \sqrt{8} }[/math]  29_{0} #14  
15.5~°  √0.𝜀^{[loweralpha 36]}  0.270~  164.5~°  √3.93~  1.982~  
2_{0} #2 
^{[loweralpha 39]}  [math]\displaystyle{ 1 / \phi\sqrt{2} }[/math]  Great rectangles in ☐ planes 
8𝝅 {30/13} #13 
28_{0} #13  
25.2~°  √0.19~  0.437~  154.8~°  √3.81~  1.952~  
3_{0} #3 
[math]\displaystyle{ \pi / 5 }[/math]  [math]\displaystyle{ 1 / \phi }[/math]  720 great decagons / 12 (3600 great rectangles) in 720 ✩ planes 
5𝝅 {15/2} #5 
[math]\displaystyle{ 4\pi / 5 }[/math]  [math]\displaystyle{ \sqrt{2+\phi} }[/math]  27_{0} #12  
36°  √0.𝚫  0.618~  144°^{[loweralpha 1]}  √3.𝚽  1.902~  
4_{0} #4−1 
[math]\displaystyle{ \sqrt{1}/\sqrt{2} }[/math]  Great rectangles in ☐ planes 
[math]\displaystyle{ \sqrt{7} / \sqrt{2} }[/math]  26_{0} #11+1  
41.4~°  √0.5  0.707~  138.6~°  √3.5  1.871~  
5_{0} #4 
[math]\displaystyle{ \sqrt{3} / \phi\sqrt{2} }[/math]  200 irregular great dodecagons^{[loweralpha 49]} / 4 (600 great rectangles) in 200 △ planes 
^{[loweralpha 48]}  [math]\displaystyle{ \phi^2 / \sqrt{2} }[/math]  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 
[math]\displaystyle{ \pi / 3 }[/math]  400 regular great hexagons^{[loweralpha 41]} / 16 (1200 great rectangles) in 200 △ planes 
4𝝅^{[loweralpha 13]} 2{10/3} #4 
[math]\displaystyle{ 2\pi / 3 }[/math]  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 
[math]\displaystyle{ 2\pi/5 }[/math]  [math]\displaystyle{ \sqrt{3\phi} }[/math]  1440 great pentagons^{[loweralpha 21]} / 12 (3600 great rectangles) in 720 ✩ planes 
8𝝅 {24/5} #9 
[math]\displaystyle{ 3\pi / 5 }[/math]  [math]\displaystyle{ \phi }[/math]  19_{0} #9  
72°  √1.𝚫  1.175~  108°  √2.𝚽  1.618~  
12_{0} #6+1 
[math]\displaystyle{ \sqrt{3} / \sqrt{2} }[/math]  1200 great digon 5cell edges^{[loweralpha 50]} / 4 (600 great rectangles) in 200 △ planes 
4𝝅^{[loweralpha 13]} {5/2} #8 
[math]\displaystyle{ \sqrt{5} / \sqrt{2} }[/math]  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 
[math]\displaystyle{ \pi / 2 }[/math]  4050 great squares^{[loweralpha 9]} / 27 in 4050 ☐ planes 
8𝝅 {30/7} #7 
[math]\displaystyle{ \pi / 2 }[/math]  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^{[loweralpha 15]} of a distinct class,^{[loweralpha 19]} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.^{[loweralpha 56]} 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.^{[loweralpha 57]} In each class of rotation,^{[loweralpha 55]} vertices rotate on a distinct kind of circular geodesic isocline^{[loweralpha 14]} which has a characteristic circumference, skew Clifford polygram^{[loweralpha 26]} and chord number, listed in the Rotation column above.^{[loweralpha 24]}
Concentric hulls
Polyhedral graph
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.^{[22]}
Constructions
The 120cell is the sixth in the sequence of 6 convex regular 4polytopes (in order of size and complexity).^{[loweralpha 3]} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the 600cell,^{[loweralpha 8]} just as the 600cell can be deconstructed into twentyfive distinct instances (or five disjoint instances) of its predecessor the 24cell,^{[loweralpha 58]} 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.^{[25]} The 120cell contains 675 distinct instances (75 disjoint instances) of the 16cell.^{[loweralpha 9]}
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.
Dual 600cells
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.
Cell rotations of inscribed duals
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.^{[27]} 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).^{[28]} 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.^{[loweralpha 8]}
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.^{[loweralpha 59]} Therefore the whole 120cell, with all ten inscribed 600cells, can be generated from just one 600cell by rotating its cells.
Augmentation
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.^{[loweralpha 60]}
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.^{[loweralpha 61]} The central cell is vertexbonded 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 [math]\displaystyle{ O(\Lambda)=W(H_4)=I }[/math] of order 120.^{[30]} The following describe [math]\displaystyle{ T }[/math] and [math]\displaystyle{ T' }[/math] 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
[math]\displaystyle{ T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{1e_1}{\sqrt{2}} & \frac{1e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{e_2e_3}{\sqrt{2}} & \frac{e_2e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{1e_2}{\sqrt{2}} & \frac{1e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{e_1e_3}{\sqrt{2}} & \frac{e_1e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{e_1e_2}{\sqrt{2}} & \frac{e_1e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{1e_3}{\sqrt{2}} & \frac{1e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix}; }[/math]
With quaternions [math]\displaystyle{ (p,q) }[/math] where [math]\displaystyle{ \bar p }[/math] is the conjugate of [math]\displaystyle{ p }[/math] and [math]\displaystyle{ [p,q]:r\rightarrow r'=prq }[/math] and [math]\displaystyle{ [p,q]^*:r\rightarrow r''=p\bar rq }[/math], then the Coxeter group [math]\displaystyle{ W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace }[/math] is the symmetry group of the 600cell and the 120cell of order 14400.
Given [math]\displaystyle{ p \in T }[/math] such that [math]\displaystyle{ \bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p }[/math] and [math]\displaystyle{ p^\dagger }[/math] as an exchange of [math]\displaystyle{ 1/\varphi \leftrightarrow \varphi }[/math] within [math]\displaystyle{ p }[/math], we can construct:
 the snub 24cell [math]\displaystyle{ S=\sum_{i=1}^4\oplus p^i T }[/math]
 the 600cell [math]\displaystyle{ I=T+S=\sum_{i=0}^4\oplus p^i T }[/math]
 the 120cell [math]\displaystyle{ J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T' }[/math]
 the alternate snub 24cell [math]\displaystyle{ S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T' }[/math]
 the dual snub 24cell = [math]\displaystyle{ T \oplus T' \oplus S' }[/math].
As a configuration
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.^{[31]}^{[32]}
[math]\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} }[/math]
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 
Visualization
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.^{[33]}
Layered stereographic projection
The cell locations lend themselves to a hyperspherical description.^{[34]} 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.
Intertwining rings
The 120cell can be partitioned into 12 disjoint 10cell great circle rings, forming a discrete/quantized Hopf fibration.^{[35]}^{[36]}^{[37]}^{[38]}^{[33]} 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.^{[39]} 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.^{[loweralpha 18]} 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.^{[loweralpha 20]} 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).
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.^{[loweralpha 17]} 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.
Perspective projections
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
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.^{[41]}
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 
File:Cell120.ogv Animated 4D rotation 
Related polyhedra and honeycombs
H_{4} polytopes
The 120cell is one of 15 regular and uniform polytopes with the same H_{4} symmetry [3,3,5]:^{[43]}
{p,3,3} polytopes
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}:
{5,3,p} polytopes
The 120cell is a part of a sequence of 4polytopes and honeycombs with dodecahedral cells:
Tetrahedrally diminished 120cell
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.^{[loweralpha 64]}
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.^{[loweralpha 38]} 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,^{[loweralpha 60]} 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.^{[loweralpha 4]}
Davis 120cell
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.
See also
 Uniform 4polytope family with [5,3,3] symmetry
 57cell – an abstract regular 4polytope constructed from 57 hemidodecahedra.
 600cell  the dual 4polytope to the 120cell
Notes
 ↑ ^{1.0} ^{1.1} ^{1.2} In the 120cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.^{[3]}
 ↑ ^{2.0} ^{2.1} The 120cell contains instances of all of the regular convex 1polytopes, 2polytopes, 3polytopes and 4polytopes, except for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which does occur. Various regular skew polygons {7} and above occur in the 120cell, notably {11},^{[loweralpha 34]} {15}^{[loweralpha 23]} and {30}.^{[loweralpha 20]}
 ↑ ^{3.0} ^{3.1} ^{3.2} The convex regular 4polytopes can be ordered by size as a measure of 4dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more 4content within the same radius. The 4simplex (5cell) is the limit smallest case, and the 120cell 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 120cell is the 600point 4polytope: sixth and last in the ascending sequence that begins with the 5point 4polytope.
 ↑ ^{4.0} ^{4.1} ^{4.2} ^{4.3} ^{4.4} ^{4.5} ^{4.6} ^{4.7} Inscribed in the unitradius 120cell are 120 disjoint regular 5cells,^{[12]} of edgelength √2.5. No regular 4polytopes except the 5cell and the 120cell contain √2.5 chords (the #8 chord).^{[loweralpha 5]} The 120cell contains 10 distinct inscribed 600cells which can be taken as 5 disjoint 600cells two different ways. Each √2.5 chord connects two vertices in disjoint 600cells, and hence in disjoint 24cells, 8cells, and 16cells. These chords and the 120cell edges occur only in the 120cell, connecting two vertices in disjoint 600cells.^{[loweralpha 22]} Corresponding polytopes of the same kind in disjoint 600cells are Clifford parallel and √2.5 apart. Each 5cell contains one vertex from each of 5 disjoint 600cells (three different ways). Each 5cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600cells together in a distinct isoclinic rotation characteristic of the 5cell.
 ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} Multiple instances of each of the regular convex 4polytopes can be inscribed in any of their larger successor 4polytopes, except for the smallest, the regular 5cell, which occurs inscribed only in the largest, the 120cell. To understand the way in which the 4polytopes nest within each other, it is necessary to carefully distinguish disjoint multiple instances from merely distinct multiple instances of inscribed 4polytopes. For example, the 600point 120cell is the convex hull of a compound of 75 8point 16cells that are completely disjoint: they share no vertices, and 75 * 8 = 600. But it is also possible to pick out 675 distinct 16cells within the 120cell, most pairs of which share some vertices, because two concentric equalradius 16cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.^{[loweralpha 9]} In 4space, any two congruent regular 4polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4simplex (the 5point regular 5cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4simplex has this property; the 16cell, and by extension any larger regular 4polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4simplex does not possess any opposing vertices (any 2vertex central axes) which might be invariant after a rotation. The 120cell contains 120 completely disjoint regular 5cells, which are its only distinct inscribed regular 5cells, but every other nesting of regular 4polytopes features some number of disjoint inscribed 4polytopes and a larger number of distinct inscribed 4polytopes.
 ↑ (Coxeter 1973) uses the greek letter 𝝓 (phi) to represent one of the three characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the golden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
 ↑ To obtain all 600 coordinates by quaternion crossmultiplication of these three 4polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24cell: (√1/2, √1/2, 0, 0).^{[9]}
 ↑ ^{8.0} ^{8.1} ^{8.2} ^{8.3} The 600 vertices of the 120cell can be partitioned into those of 5 disjoint inscribed 120vertex 600cells in two different ways.^{[29]} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells. The 120cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600cells.
 ↑ ^{9.0} ^{9.1} ^{9.2} ^{9.3} ^{9.4} ^{9.5} ^{9.6} ^{9.7} The 120cell has 600 vertices distributed symmetrically on the surface of a 3sphere in fourdimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 rays [or axes] of the 120cell. 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. The rays and bases constitute a geometric configuration, which in the language of configurations is written as 300_{9}675_{4} to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.^{[26]} Each basis corresponds to a distinct 16cell containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16cells containing all 600 vertices of the 120cell can be selected from the 675 distinct 16cells.^{[loweralpha 5]}
 ↑ ^{10.0} ^{10.1} ^{10.2} ^{10.3} ^{10.4} ^{10.5} The 120cell can be constructed as a compound of 5 disjoint 600cells,^{[loweralpha 8]} or 25 disjoint 24cells, or 75 disjoint 16cells, or 120 disjoint 5cells. Except in the case of the 120 5cells,^{[loweralpha 5]} these are not counts of all the distinct regular 4polytopes which can be found inscribed in the 120cell, only the counts of completely disjoint inscribed 4polytopes which when compounded form the convex hull of the 120cell. The 120cell contains 10 distinct 600cells, 225 distinct 24cells, and 675 distinct 16cells.^{[loweralpha 9]}
 ↑ ^{11.0} ^{11.1} ^{11.2} The edges and 8𝝅 characteristic rotations of the 16cell lie in the great square ☐ central planes. More generally, rotations of this type are a consequence of the symmetry group [math]\displaystyle{ B 4 }[/math]. The edges and 5𝝅 characteristic rotations of the 600cell lie in the great pentagon ✩ (great decagon) central planes. More generally, rotations of this type are a consequence of the symmetry group [math]\displaystyle{ H 4 }[/math]. The edges and 4𝝅 characteristic rotations^{[loweralpha 13]} of the other regular 4polytopes, the regular 5cell, the 8cell hypercube, the 24cell, and the 120cell,^{[loweralpha 23]} all lie in the great triangle △ (great hexagon) central planes.^{[loweralpha 18]} Collectively these rotations involve all four symmetry groups [math]\displaystyle{ A 4 }[/math], [math]\displaystyle{ B 4 }[/math], [math]\displaystyle{ F 4 }[/math] and [math]\displaystyle{ H 4 }[/math].
 ↑ The finite length of an ordinary great circle is always 2𝝅r of course, but the isoclines of each discrete nonsimple rotation have their own characteristic circumferential length, which is in every case greater than 2𝝅r.^{[loweralpha 11]}
 ↑ ^{13.0} ^{13.1} ^{13.2} ^{13.3} ^{13.4} All 3sphere isoclines^{[loweralpha 14]} of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference 2𝝅; simple rotations take place on 2𝝅 isoclines. Double rotations may have isoclines of up to 8𝝅 circumference. Because the characteristic rotations of several regular 4polytopes take place in the same invariant planes (the 24cell's hexagonal planes), those rotations all have congruent isoclines of 4𝝅 circumference. The regular 4polytopes which rotate on 4𝝅 isoclines characteristically (when they are rotating in the isoclinic invariant planes containing their edges) are the 5cell, the 8cell, the 24cell and the 120cell.
 ↑ ^{14.0} ^{14.1} ^{14.2} ^{14.3} ^{14.4} ^{14.5} ^{14.6} An isocline is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3dimensional space of the 4polytope's boundary surface it is a straight line, a geodesic. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are linked and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3sphere's natural curvature, within which curved space they are finite (closed) straight line segments.^{[loweralpha 12]} To avoid confusion, we always refer to an isocline as such, and reserve the term great circle for an ordinary great circle in the plane.^{[loweralpha 13]}
 ↑ ^{15.0} ^{15.1} ^{15.2} ^{15.3} ^{15.4} ^{15.5} An isoclinic^{[loweralpha 16]} rotation is an equirotationangled double rotation in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arcangles (chord lengths), its rotation angle and its isocline angle.^{[loweralpha 19]} In each incremental rotation step from vertex to neighboring vertex, each rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle. Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.^{[loweralpha 54]} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two adjacent great circle polygons (the distance between their corresponding vertices in the rotation).
 ↑ ^{16.0} ^{16.1} ^{16.2} ^{16.3} ^{16.4} ^{16.5} Two angles are required to specify the separation between two planes in 4space.^{[11]} If the two angles are identical, the two planes are called isoclinic (also Clifford parallel) and they intersect in a single point. In double rotations, points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a helical smooth curve called an isocline^{[loweralpha 14]} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a Clifford displacement). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.^{[loweralpha 15]}
 ↑ ^{17.0} ^{17.1} ^{17.2} ^{17.3} ^{17.4} ^{17.5} ^{17.6} Cite error: Invalid
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 ↑ ^{18.0} ^{18.1} ^{18.2} ^{18.3} ^{18.4} ^{18.5} ^{18.6} ^{18.7} ^{18.8} ^{18.9} The 120cell has an irregular dodecagon {12} great circle polygon of 6 edges (#1 chords marked 𝜁) alternating with 6 dodecahedron celldiameters (#4 chords).^{[loweralpha 37]} The irregular great dodecagon contains two irregular great hexagons (red) inscribed in alternate positions.^{[loweralpha 17]} Two regular great hexagons with edges of a third size (√1, the #5 chord) are also inscribed in the dodecagon.^{[loweralpha 41]} The twelve regular hexagon edges (#5 chords), the six celldiameter edges of the dodecagon (#4 chords), and the six 120cell edges of the dodecagon (#1 chords), are all chords of the same great circle, but the other 24 zigzag edges (#1 chords, not shown) that bridge the six #4 edges of the dodecagon do not lie in this great circle plane. The 120cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of dodecagon planes. The 120cell contains 200 such {12} central planes (100 completely orthogonal pairs), the same 200 central planes each containing a hexagon that are found in each of the 10 inscribed 600cells.^{[loweralpha 40]}
 ↑ ^{19.0} ^{19.1} ^{19.2} ^{19.3} Every class of discrete isoclinic rotation^{[loweralpha 15]} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The characteristic isoclinic rotation of a 4polytope is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each Hopf fibration of the edge planes). If the edges of the 4polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arcangle (the edge chord is simply the rotation chord). But in a regular 4polytope with a tetrahedral vertex figure^{[loweralpha 27]} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120cell are edges of an irregular great dodecagon which also has #4 chord edges.^{[loweralpha 18]} In such a 4polytope, the rotation angle is not the edge arcangle; in fact it is not necessarily the arc of any vertex chord.^{[loweralpha 47]}
 ↑ ^{20.0} ^{20.1} ^{20.2} ^{20.3} ^{20.4} ^{20.5} ^{20.6} The 120cell and 600cell both have 30gon Petrie polygons.^{[loweralpha 29]} They are two distinct skew 30gon helices, composed of 30 120cell edges (#1 chords) and 30 600cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0gon great circle axis. The 120cell's Petrie helix winds closer to the axis than the 600cell's Petrie helix does, because its 30 edges are shorter than the 600cell's 30 edges (and they zigzag at less acute angles). A dual pair^{[loweralpha 29]} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.^{[loweralpha 33]} The 120cell Petrie helix (versus the 600cell Petrie helix) twists around the 0gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew {30/9}=3{10/3} polygram (versus a skew {30/11} polygram).^{[loweralpha 34]}
 ↑ ^{21.0} ^{21.1} ^{21.2} Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120cell^{[loweralpha 4]} are 6 great pentagonsCite error: Invalid
<ref>
tag; refs with no name must have content in which the 6 pentagrams (regular 5cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5cell actually has vertices in 5 different decagonpentagon central planes in 5 completely disjoint 600cells.  ↑ The 120 regular 5cells are completely disjoint. Each 5cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600cells together. But the vertices of two disjoint 5cells are not linked by 5cell edges, so each {5/2} pentagram isocline of 5 #8 chords is confined to a single 5cell, and there are no other circuits of 5cell edges (#8 chords) in the 120cell.
 ↑ ^{23.0} ^{23.1} ^{23.2} ^{23.3} ^{23.4} ^{23.5} ^{23.6} ^{23.7} ^{23.8} ^{23.9} thumb
 ↑ ^{24.0} ^{24.1} The 120cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.^{[21]}
 ↑ Cite error: Invalid
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tag; no text was provided for refs namedpentadecagram isoclines
 ↑ ^{26.0} ^{26.1} ^{26.2} The chordpath of an isocline^{[loweralpha 14]} may be called the 4polytope's Clifford polygon, as it is the skew polygram shape of the rotational circles traversed by the 4polytope's vertices in its characteristic Clifford displacement.^{[loweralpha 16]}
 ↑ ^{27.0} ^{27.1} The 5cell, 8cell and 120cell all have tetrahedral vertex figures. In a 4polytope with a tetrahedral vertex figure, a 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. In the 120cell the 30edge circumferential path along edges follows a zigzag skew Petrie polygon, which is not a great circle. However, there exists a 15chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical isocline^{[loweralpha 14]} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.^{[loweralpha 25]} The skew chord set of an isocline is called its Clifford polygon.^{[loweralpha 26]}
 ↑ ^{28.0} ^{28.1} The 30edge circumference of the 120cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4polytope is a zigzag helix spiraling through the curved 3space of the 4polytope's surface.^{[loweralpha 31]} The 15 numbered chords of the 120cell occur as the distance between two vertices in that 30vertex helical ring.^{[loweralpha 32]} Those 15 distinct Pythagorean distances through 4space range from the 120cell edgelength which links any two nearest vertices in the ring (the #1 chord), to the 120cell axislength (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).
 ↑ ^{29.0} ^{29.1} ^{29.2} The regular skew 30gon is the Petrie polygon of the 600cell and its dual the 120cell. The Petrie polygons of the 120cell occur in the 600cell as duals of the 30cell Boerdijk–Coxeter helix rings (the Petrie polygons of the 600cell):^{[loweralpha 34]} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120cell partitions into 20 nonintersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120cell (just as their 20 dual 30cell rings are a discrete fibration of the 600cell).^{[loweralpha 20]}
 ↑ ^{30.0} ^{30.1} ^{30.2} ^{30.3} The 120cell contains 80 distinct 30gon Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30gon Petrie polygons.^{[loweralpha 29]} The Petrie 30gon twists around its 0gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound triacontagram {30/9}=3{10/3} of 600cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.^{[loweralpha 20]} The {30/9}gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30gon (with its #1 chord edges).
 ↑ The Petrie polygon of a 3polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edgebonded faces bent into a ring. Within that circular strip of edgebonded triangles (10 in the case of the icosahedron) the Petrie polygon can be picked out as a skew polygon of edges zigzagging (not circling) through the 2space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4polytope (polychoron) with tetrahedral cells (e.g. a 600cell) can be seen as a linear helix of facebonded cells bent into a ring: a Boerdijk–Coxeter helix ring. Within that circular helix of facebonded tetrahedra (30 in the case of the 600cell) the skew Petrie polygon can be picked out as a helix of edges zigzagging (not circling) through the 3space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.
 ↑ ^{32.0} ^{32.1} ^{32.2} ^{32.3} ^{32.4} ^{32.5} ^{32.6} ^{32.7} The 120cell itself contains more chords than the 15 chords numbered #1  #15, but the additional chords occur only in the interior of 120cell, not as edges of any of the six regular convex 4polytopes or their characteristic great circle rings. The 15 major chords are so numbered because the #n chord connects two vertices which are n edge lengths apart on a Petrie polygon. There are 30 distinct 4space chordal distances between vertices of the 120cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we name the 15 unnumbered minor chords by their arcangles, e.g. 41.4~° which, with length √0.5, falls between the #3 and #4 chords.
 ↑ ^{33.0} ^{33.1} "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 _{0}𝑹, _{1}𝑹, _{2}𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈_{0}, 𝐈𝐈_{1}, 𝐈𝐈_{2}, ... of the original polytope."^{[15]}
 ↑ ^{34.0} ^{34.1} ^{34.2} ^{34.3} 180px
 ↑ ^{35.0} ^{35.1} ^{35.2} The characteristic isoclinic rotation of the 120cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation^{[loweralpha 15]} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,^{[loweralpha 14]} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arclength.^{[loweralpha 47]}
 ↑ ^{36.0} ^{36.1} 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 = 𝚽^{2} ≈ 0.382
𝜀 = √𝚫^{2}/2 ≈ 0.073
and the 120cell edgelength 1/φ^{2}√2 is:
𝛇 = √𝜀 ≈ 0.270
For example:
𝛇 = √0.𝜀 = √0.073~ ≈ 0.270  ↑ ^{37.0} ^{37.1} ^{37.2} ^{37.3} In the dodecahedral cell of the unitradius 120cell, the length of the edge (the #1 chord of the 120cell) is 1/φ^{2}√2 ≈ 0.270. Eight orange vertices lie at the Cartesian coordinates (±φ^{3}√8, ±φ^{3}√8, ±φ^{3}√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 120cell). 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 (600cell edges, and the #3 chord of the 120cell). The diameter of the dodecahedron is √3/φ√2 ≈ 0.757 (the cube diagonal, and the #4 chord of the 120cell).
 ↑ ^{38.0} ^{38.1} The face pentagon diagonal (the #2 chord) is in the golden ratio φ ≈ 1.618 to the face pentagon edge (the 120cell edge, the #1 chord).^{[loweralpha 37]}
 ↑ ^{39.0} ^{39.1} ^{39.2} The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120cell's tetrahedral vertex figure, the second section of the 120cell beginning with a vertex, denoted 1_{0}. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120cell's pentagon faces.^{[loweralpha 38]}
 ↑ ^{40.0} ^{40.1} ^{40.2} The 120cell contains ten 600cells which can be partitioned into five completely disjoint 600cells two different ways.^{[loweralpha 8]} All ten 600cells occupy the same set of 200 irregular great dodecagon central planes.^{[loweralpha 18]} There are exactly 400 regular hexagons in the 120cell (two in each dodecagon central plane), and each of the ten 600cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600cell is disjoint from 4 other 600cells, and shares hexagons with 5 other 600cells.^{[loweralpha 63]} Each disjoint pair of 600cells occupies the opposing pair of disjoint great hexagons in every dodecagon central plane. Each nondisjoint pair of 600cells intersects in 16 hexagons that comprise a 24cell. The 120cell contains 9 times as many distinct 24cells (225) as disjoint 24cells (25).^{[loweralpha 9]} Each 24cell occurs in 9 600cells, is absent from just one 600cell, and is shared by two 600cells.
 ↑ ^{41.0} ^{41.1} ^{41.2} ^{41.3} Each great hexagon edge is the axis of a zigzag of 5 120cell edges. The 120cell's Petrie polygon is a helical zigzag of 30 120cell edges, spiraling around a 0gon great circle axis that does not intersect any vertices.^{[loweralpha 20]} There are 5 great hexagons inscribed in each Petrie polygon, in five different central planes.^{[loweralpha 40]}
 ↑ The Petrie polygon of the 5cell is the pentagon (5gon), and the Petrie polygon of the 120cell is the triacontagon (30gon).^{[loweralpha 30]} Each 120cell Petrie 30gon lies completely orthogonal to six 5cell Petrie 5gons, which belong to six of the 120 disjoint regular 5cells inscribed in the 120cell.^{[loweralpha 4]}
 ↑ The sum of the squared lengths of all the distinct chords of any regular convex npolytope of unit radius is the square of the number of vertices.^{[16]}
 ↑ Dodecahedra emerge as visible features in the 120cell, but they also occur in the 600cell as interior polytopes.^{[18]}
 ↑ In the curved 3dimensional space of the 120cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the apex vertex and its surrounding base polyhedron form a polyhedral pyramid. The characteristic chord is radial around the apex, as the pyramid's lateral edges.
 ↑ In the 120cell's isoclinic rotations the rotation arcangle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120cell isoclinic rotation by 12° will take the great polygon in every central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by two 120cell edges (#1 chords of arclength 15.5~°). The 12° rotation angle is not the arc of any vertextovertex chord in the 120cell. It occurs only as the two equal angles between adjacent Clifford parallel central planes,^{[loweralpha 16]} and it is the separation between adjacent rotation planes in all the 120cell's various isoclinic rotations (not only in its characteristic rotation).
 ↑ ^{47.0} ^{47.1} ^{47.2} The isocline chord of the 120cell's characteristic rotation^{[loweralpha 23]} is the #4 chord of 44.5~° arcangle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation each vertex moves to another vertex 4 edgelengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline)^{[loweralpha 14]} does not intersect any nearer vertices.
 ↑ ^{48.0} ^{48.1} Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.^{[loweralpha 15]} The #4 chord^{[loweralpha 35]} bridge is significant in an isoclinic rotation in regular great hexagons (the 24cell's characteristic rotation), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120cell's characteristic rotation (in irregular great hexagons).^{[loweralpha 23]} In each 12° arc^{[loweralpha 47]} of the 24cell's characteristic rotation of the 120cell, every regular great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.
 ↑ This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The blue central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a √4 axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown).^{[loweralpha 18]} Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.^{[loweralpha 48]}
 ↑ The regular 5cell has only digon central planes intersecting two vertices. The 120cell with 120 inscribed regular 5cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5cells of length √2.5. Three disjoint rectangles occur in one {12} central plane, where the six #8 √2.5 chords belong to six disjoint 5cells. The 12_{0} sections and 18_{0} sections are regular tetrahedra of edge length √2.5, the cells of regular 5cells. The regular 5cells' ten triangle faces lie in those sections; each of a face's three √2.5 edges lies in a different {12} central plane.
 ↑ ^{51.0} ^{51.1} Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4polytope, in planes that do not intersect except at one point, the common center of the two linked circles.
 ↑ The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.^{[loweralpha 51]}
 ↑ The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4polytope, all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.^{[20]} The corresponding vertices of the two completely orthogonal great polygons are √4 (180°) apart; the great polygons (Clifford parallel polytopes) are √4 (180°) apart; but the two completely orthogonal planes are 90° apart, in the two orthogonal angles that separate them.^{[loweralpha 16]} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different orientation (axes swapped): it has been turned "upside down" on the surface of the 4polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.
 ↑ It is easiest to visualize this incorrectly, because the completely orthogonal great circles are Clifford parallel and do not intersect. An invariant plane tilts sideways in an orthogonal central plane which is not its completely orthogonal plane, but Clifford parallel to it. It rotates with its completely orthogonal plane, but not in it. It is Clifford parallel to its completely orthogonal plane, and does not intersect it; the plane that it rotates in is orthogonal to both completely orthogonal planes and intersects them both.^{[loweralpha 52]} In the 120cell's characteristic rotation,^{[loweralpha 23]} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.^{[loweralpha 53]}
 ↑ ^{55.0} ^{55.1} Rotations in 4dimensional Euclidean space are defined by at least one pair of completely orthogonal^{[loweralpha 51]} central planes of rotation which are invariant, which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic^{[loweralpha 16]} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually Clifford parallel. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.^{[loweralpha 19]} It has multiple distinct left (and right) rotation instances called fibrations, which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular fibers, the great circle polygons in their invariant planes.
 ↑ In the 120cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.^{[loweralpha 55]}
 ↑ Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons / 4".
 ↑ The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600cell do not, so each of the 10 must belong to a different 600cell.
 ↑ ^{60.0} ^{60.1} Each 120cell vertex figure is actually a low tetrahedral pyramid, an irregular 5cell with a regular tetrahedron base.
 ↑ As we saw in the 600cell, these 12 tetrahedra belong (in pairs) to the 6 icosahedral clusters of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.
 ↑ A 24cell contains 16 hexagons. In the 600cell, with 25 24cells, each 24cell is disjoint from 8 24cells and intersects each of the other 16 24cells in six vertices that form a hexagon.^{[40]} A 600cell contains 25・16/2 = 200 such hexagons.
 ↑ Each regular great hexagon is shared by two 24cells in the same 600cell,^{[loweralpha 62]} and each 24cell is shared by two 600cells.^{[loweralpha 58]} Each regular hexagon is shared by four 600cells.
 ↑ The diminishment of the 600point 120cell to a 480point 4polytope by removal of one if its 600cells is analogous to the diminishment of the 120point 600cell by removal of one of its 5 disjoint inscribed 24cells, creating the 96point snub 24cell. Similarly, the 8cell tesseract can be seen as a 16point diminished 24cell from which one 8point 16cell has been removed.
Citations
 ↑ N.W. Johnson: Geometries and Transformations, (2018) ISBN:9781107103405 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
 ↑ Matila Ghyka, The Geometry of Art and Life (1977), p.68
 ↑ ^{3.0} ^{3.1} Coxeter 1973, pp. 292293, Table I(ii); "120cell".
 ↑ Dechant 2021, p. 20, Remark 5.7, explains why not.^{[loweralpha 5]}
 ↑ 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 CartanDieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600cell) 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 0387953361
 ↑ ^{7.0} ^{7.1} Stillwell 2001.
 ↑ Coxeter 1973, pp. 156157, §8.7 Cartesian coordinates.
 ↑ ^{9.0} ^{9.1} Mamone, Pileio & Levitt 2010, p. 1442, Table 3.
 ↑ Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Fourdimensional 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 [math]\displaystyle{ \left(w,x,y,z\right)_1 }[/math] and [math]\displaystyle{ \left(w,x,y,z\right)_2 }[/math] according to
[math]\displaystyle{ \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} }[/math]  ↑ Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between kdimensional subspaces.)".
 ↑ Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.
 ↑ Mamone, Pileio & Levitt 2010, pp. 14381439, §4.5 Regular Convex 4Polytopes, Table 2, Symmetry group 𝛨_{4}; the 120cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct isoclinic rotations.^{[loweralpha 24]}
 ↑ Coxeter 1973, pp. 300301, Table V:(v) Simplified sections of {5,3,3} (edge 2φ^{−2}√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 nonpoint sections labelled 1_{0} − 16_{0}, polyhedra whose successively increasing "radii" on the 3sphere (in column 2la) are the following chords in our notation:^{[loweralpha 32]} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column a for (30−i)_{0}) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeterlisted chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edgelengths of the polyhedral sections (in column 2lb, which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edgelengths of irregular polyhedral sections are not given).
 ↑ Coxeter 1973, p. 147, §8.1 The simple truncations of the general regular 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.
 ↑ Coxeter 1973, pp. 300301, Table V:(v) Simplified sections of {5,3,3} (edge 2φ^{−2}√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 nonpoint sections labelled 1_{0} − 16_{0}, but 14_{0} and 16_{0} are congruent opposing sections and 15_{0} opposes itself; there are 29 nonpoint sections, denoted 1_{0} − 29_{0}, in 15 opposing pairs.
 ↑ Kim & Rote 2016, pp. 810, Relations to Clifford Parallelism.
 ↑ Mamone, Pileio & Levitt 2010, §4.5 Regular Convex 4Polytopes, Table 2.
 ↑ Carlo H. Séquin (July 2005). Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes. Mathartfun.com. pp. 463–472. ISBN 9780966520163. https://archive.bridgesmathart.org/2005/bridges2005463.html#gsc.tab=0. Retrieved March 13, 2023.
 ↑ van Ittersum 2020, p. 435, §4.3.5 The two 600cells circumscribing a 24cell.
 ↑ 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. 34, §2 Geometry of the 120cell: rays and bases.
 ↑ Sullivan 1991, pp. 45, 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 mirrorimage) is said to be chiral."
 ↑ Waegell & Aravind 2014, pp. 56.
 ↑ Koca, AlAjmi & Ozdes Koca 2011, pp. 986988, 6. Dual of the snub 24cell.
 ↑ Coxeter 1973, §1.8 Configurations.
 ↑ Coxeter 1991, p. 117.
 ↑ ^{33.0} ^{33.1} Sullivan 1991, p. 15, Other Properties of the 120cell.
 ↑ Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.
 ↑ Coxeter 1970, pp. 1923, §9. The 120cell and the 600cell.
 ↑ Schleimer & Segerman 2013, pp. 1618, §6.2. Rings of dodecahedra.
 ↑ Banchoff 2013.
 ↑ Zamboj 2021, pp. 612, §2 Mathematical background.
 ↑ Zamboj 2021, pp. 2329, §5 Hopf tori corresponding to circles on B^{2}.
 ↑ Denney et al. 2020, p. 438.
 ↑ Chilton 1964.
 ↑ Dechant 2021, pp. 1820, 6. The Coxeter Plane.
 ↑ Denney et al. 2020.
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External links
 Weisstein, Eric W.. "120Cell". http://mathworld.wolfram.com/120Cell.html.
 Olshevsky, George. "Hecatonicosachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#hecatonicosachoron.
 Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5x  hi". https://bendwavy.org/klitzing/dimensions/polychora.htm.
 Der 120Zeller (120cell) Marco Möller's Regular polytopes in R^{4} (German)
 120cell explorer – A free interactive program that allows you to learn about a number of the 120cell symmetries. The 120cell is projected to 3 dimensions and then rendered using OpenGL.
 Construction of the HyperDodecahedron
 YouTube animation of the construction of the 120cell Gian Marco Todesco.
Original source: https://en.wikipedia.org/wiki/120cell.
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