120-cell
In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C120, dodecaplex, hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.
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. As the sixth and largest regular convex 4-polytope, it contains inscribed instances of its four predecessors. 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 mathematics 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 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 and of the following:
| 24 | 24-cell | 600-point 120-cell | |
| 64 | 600-point 120-cell | ||
| 64 | 600-point 120-cell | ||
| 64 | 600-point 120-cell | ||
| 96 | Snub 24-cell | 600-point 120-cell | |
| 96 | Snub 24-cell | 600-point 120-cell | |
| 192 | 600-point 120-cell |
where φ 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 and in the left column below:
The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The 120-cell is the convex hull of 5 disjoint 600-cells. Each 600-cell is the convex hull of 5 disjoint 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 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the 120-cell is the convex hull of 120 disjoint 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 zig-zag skew polygon.
Since the 120-cell has a circumference of 30 edges, it has 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 is marked with a triangle △, square ☐, phi symbol ? 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 ? 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.
=15
!colspan=6|Squared lengths total
!
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 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. 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 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 central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 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.The 120-cell's irregular dodecagon great circle polygon has 6 short edges alternating with 6 longer dodecahedron cell-diameters. Inscribed in the irregular great dodecagon are two irregular great hexagons in alternate positions. Two regular great hexagons with edges of a third size are also inscribed in the dodecagon. The 120-cell'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 200 dodecagon planes. They occur as 100 completely orthogonal pairs, and they are the same 200 central planes each containing a hexagon that are found in each of the 10 inscribed 600-cells.
There are exactly 400 regular hexagons in the 120-cell, and each of the ten 600-cells contains its own distinct subset of 200 of them. Each 600-cell 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 600-cell is disjoint from 4 other 600-cells, and shares regular hexagons with 5 other 600-cells. Each disjoint pair of 600-cells occupies the opposing pair of disjoint regular hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells as disjoint 24-cells. Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells.