In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoronN.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249 and hecatonicosahedroid.Matila Ghyka, The Geometry of Art and Life (1977), p.68
The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. Together they form 720 pentagonal faces, 1200 edges, and 600 vertices. It is the 4-dimensional analogue of the regular dodecahedron, since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the dodecaplex has 120 dodecahedral facets, with 3 around each edge. Its dual polytope is the 600-cell.
Geometry The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above). As the sixth and largest regular convex 4-polytope, it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5-cell, which is not found in any of the others.
The 120-cell 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. That is why Stillwell titled his paper on the 4-polytopes and the history of mathematicsMathematics and Its History, John Stillwell, 1989, 3rd edition 2010, of more than 3 dimensions The Story of the 120-cell.
Cartesian coordinates Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen.
√8 radius coordinates The 120-cell with long radius = 2 ≈ 2.828 has edge length 4−2φ = 3− ≈ 0.764.
In this frame of reference, its 600 vertex coordinates are the {permutations} and of the following: 24 ({0, 0, ±2, ±2}) 24-cell 600-point 120-cell64 ({±φ, ±φ, ±φ, ±φ−2})64 ({±1, ±1, ±1, ±})64 ({±φ−1, ±φ−1, ±φ−1, ±φ2})96 ([0, ±φ−1, ±φ, ±]) Snub 24-cell96 ([0, ±φ−2, ±1, ±φ2]) Snub 24-cell192 ([±φ−1, ±1, ±φ, ±2])
where φ (also called 𝝉) is the golden ratio, ≈ 1.618.
Unit radius coordinates The unit-radius 120-cell has edge length ≈ 0.270.
In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates are the {permutations} and in the left column below: 1208({±1, 0, 0, 0})16-cell24-cell600-cell120-cell16({±1, ±1, ±1, ±1}) / 2Tesseract96([0, ±φ−1, ±1, ±φ]) / 2Snub 24-cell480Diminished 120-cell5-point 5-cell24-cell600-cell32([±φ, ±φ, ±φ, ±φ−2]) / (1, 0, 0, 0) (−1,,,) / 4 (−1,−,−,) / 4 (−1,−,,−) / 4 (−1,,−,−) / 4({±, ±, 0, 0})({±1, 0, 0, 0}) ({±1, ±1, ±1, ±1}) / 2 ([0, ±φ−1, ±1, ±φ]) / 232([±1, ±1, ±1, ±]) / 32([±φ−1, ±φ−1, ±φ−1, ±φ2]) / 96([0, ±φ−1, ±φ, ±]) / 96([0, ±φ−2, ±1, ±φ2]) / 192([±φ−1, ±1, ±φ, ±2]) / The unit-radius coordinates of uniform convex 4-polytopes are related by quaternion multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible quaternion products of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).
The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.
Chords
The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells. These two additional chords give the 120-cell its characteristic isoclinic rotation, in addition to all the rotations of the other regular 4-polytopes which it inherits. They also give the 120-cell a characteristic great circle polygon: an irregular great hexagon in which three 120-cell edges alternate with three 5-cell edges.
The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the 5-cell and the 8-cell tesseract, they form zig-zag Petrie polygons instead. The 120-cell's Petrie polygon is a triacontagon {30} zig-zag skew polygon.
Since the 120-cell has a circumference of 30 edges, it has at least 15 distinct chord lengths, ranging from its edge length to its diameter. Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.
The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point snub 24-cell and the 480-point diminished 120-cell.
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 16-cell, great hexagons and great triangles △ characteristic of the 24-cell, great decagons and great pentagons 𝜙 characteristic of the 600-cell, and skew pentagrams ✩ characteristic of the 5-cell which circle through a set of central planes and form face polygons but not great polygons.
Major chords of the 120-cell and its inscribed 4-polytopesInscribed5-cell16-cell8-cell24-cellSnub600-cellDimin120-cellVertices58162496120480600Edges1024329643272012001200Edge chord#8#7#5#5#3#3#1#1Isocline chord#8#15#10#10#5#5#4#4Clifford polygon{5/2}{8/3}{6/2}{15/2}{15/4}ChordArcEdge#1△30120-cell edge1200{3,3}15.5~°0.270~#2☐15face diagonal36002{3,4}25.2~°0.437~#3𝜙10𝝅/5great decagon 72072002{3,5}36°0.618~#4△cell diameter1200{3,3}44.5~°0.757~#5△6𝝅/3great hexagon3296120024004{4,3}60°1#6𝜙52𝝅/5great pentagon72072002{3,5}72°1.175~#7☐𝝅/2great square244872180090009{3,4}90°1.414~#8✩5-cell edge107201200{3,3}104.5~°1.581~#9𝜙3𝝅/5golden section 72072002{3,5}108°1.618~#10△32𝝅/3great triangle3296120024004{4,3}120°1.732~#11✩{30/11}-gram1200{3,3}135.5~°1.851~#12𝜙4𝝅/5great pent diag72072002{3,5}144°1.902~#13✩{30/13}-gram36002{3,4}154.8~°1.952~#14△{30/14}=2{15/7}1200{3,3}164.5~°1.982~#15△☐𝜙2𝝅diameter48124860240300180°2Squared lengths total256425657614400360000
The annotated chord table is a complete bill of materials for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table.
The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the #3 chord row, the 600-cell's 72 great decagons contain 720 #3 chords in all.
The integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of disjoint 24-cells (25 * 24 vertices = 600 vertices).
The integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains distinct 24-cells which share components.
The integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the #3 chord row, #3 chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral vertex figure 2{3,5}. In total major chords of 15 distinct lengths meet at each vertex of the 120-cell.
Relationships among interior polytopes The 120-cell is the compound of all five of the other regular convex 4-polytopes. All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell. It is a four-dimensional jigsaw puzzle in which all those polytopes are the parts. Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.
The regular 1-polytope occurs in only 15 distinct lengths in any of the component polytopes of the 120-cell. By Alexandrov's uniqueness theorem, convex polyhedra with shapes distinct from each other also have distinct metric spaces of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords.
Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four , , and are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built from them.
An additional 4 of the 15 chords are required to build the 600-cell. The four are square roots of irrational fractions that are functions of . The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra.
All 15 chords, and 15 other distinct chordal distances enumerated below, occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are The relationships between the regular 5-cell (the simplex regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell. The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells. Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.
Compound of five 600-cells
The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways. As a consequence of being a compound of five disjoint 600-cells, the 120-cell has 200 irregular great dodecagon {12} central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 {12} central planes originate as the compounds of the hexagonal central planes of the 25 disjoint inscribed 24-cells and the digon central planes of the 120 disjoint inscribed regular 5-cells; they contain all the 24-cell and 5-cell edges, and also the 120-cell edges. Thus the edges and characteristic rotations of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in these same 200 rotation planes. Each of the ten 600-cells occupies the entire set of 200 planes.