120-cell | |
---|---|

Type | Convex regular 4-polytope |

Schläfli symbol | {5,3,3} |

Coxeter diagram | |

Cells | 120 {5,3} |

Faces | 720 {5} |

Edges | 1200 |

Vertices | 600 |

Vertex figure | tetrahedron |

Petrie polygon | 30-gon |

Coxeter group | H_{4}, [3,3,5] |

Dual | 600-cell |

Properties | convex, isogonal, isotoxal, isohedral |

Uniform index | 32 |

In geometry, the **120-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a **C _{120}**,

- Geometry
- Cartesian coordinates
- Polyhedral graph
- Constructions
- As a configuration
- Visualization
- Layered stereographic projection
- Intertwining rings
- Other great circle constructs
- Projections
- Orthogonal projections
- Perspective projections
- Related polyhedra and honeycombs
- H4 polytopes
- {p,3,3} polytopes
- {5,3,p} polytopes
- Davis 120-cell
- See also
- Notes
- Citations
- References
- External links

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.^{ [lower-alpha 1] } Its dual polytope is the 600-cell.

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,^{ [lower-alpha 2] } 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.^{ [4] } The 120-cell is a four-dimensional Swiss Army knife: it contains one of everything.

It is daunting but instructive to study the 120-cell, because it contains examples of *every* relationship among *all* the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit. That is why Stillwell titled his paper on the 4-polytopes and the history of mathematics^{ [5] } of more than 3 dimensions *The Story of the 120-cell*.^{ [6] }

Regular convex 4-polytopes | |||||||
---|---|---|---|---|---|---|---|

Symmetry group | A_{4} | B_{4} | F_{4} | H_{4} | |||

Name | 5-cell Hyper-tetrahedron | 16-cell Hyper-octahedron | 8-cell Hyper-cube | 24-cell
| 600-cell Hyper-icosahedron | 120-cell Hyper-dodecahedron | |

Schläfli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} | |

Coxeter mirrors | |||||||

Mirror dihedrals | 𝝅/2𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2 | 𝝅/2𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2 | 𝝅/2𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2 | 𝝅/2𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2 | 𝝅/2𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2 | 𝝅/2𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2 | |

Graph | |||||||

Vertices | 5 | 8 | 16 | 24 | 120 | 600 | |

Edges | 10 | 24 | 32 | 96 | 720 | 1200 | |

Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons | |

Cells | 5 tetrahedra | 16 tetrahedra | 8 cubes | 24 octahedra | 600 tetrahedra | 120 dodecahedra | |

Tori | 1 5-tetrahedron | 2 8-tetrahedron | 2 4-cube | 4 6-octahedron | 20 30-tetrahedron | 12 10-dodecahedron | |

Inscribed | 120 in 120-cell | 675 in 120-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells | |

Great polygons | 2 𝝅/2 squares x 3 | 4 𝝅/2 rectangles x 3 | 4 𝝅/3 hexagons x 4 | 12 𝝅/5 decagons x 6 | 50 𝝅/15 dodecagons x 4 | ||

Petrie polygons | 1 pentagon | 1 octagon | 2 octagons | 2 dodecagons | 4 30-gons | 20 30-gons | |

Isocline polygons | 1 {8/2}=2{4} x {8/2}=2{4} | 2 {8/2}=2{4} x {8/2}=2{4} | 2 {12/2}=2{6} x {12/6}=6{2} | 4 {30/2}=2{15} x 30{0} | 20 {30/2}=2{15} x 30{0} | ||

Long radius | |||||||

Edge length | |||||||

Short radius | |||||||

Area | |||||||

Volume | |||||||

4-Content |

The 600 vertices of a 120-cell with an edge length of 2/φ^{2} = 3−√5 and a center-to-vertex radius of √8 = 2 √2 include all permutations of:

- (0, 0, ±2, ±2)
- (±1, ±1, ±1, ±√5)
- (±φ
^{−2}, ±φ, ±φ, ±φ) - (±φ
^{−1}, ±φ^{−1}, ±φ^{−1}, ±φ^{2})

and all even permutations of

- (0, ±φ
^{−2}, ±1, ±φ^{2}) - (0, ±φ
^{−1}, ±φ, ±√5) - (±φ
^{−1}, ±1, ±φ, ±2)

where φ is the golden ratio, 1 + √5/2.^{ [7] }

Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter is 15, connecting each vertex to its coordinate-negation, at a Euclidean distance of 4√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 2/φ^{2}, with a multiplicity of 4, to 4, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582.

The vertices of the 120-cell polyhedral graph are 3-colorable.

It has not been published whether the graph is Hamiltonian or Eulerian or both or neither.

The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^{ [lower-alpha 2] } It can be deconstructed into ten distinct instances (or five disjoint instances) of its immediate predecessor (and dual) the 600-cell,^{ [lower-alpha 3] } just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its immediate predecessor the 24-cell,^{ [lower-alpha 4] } the 24-cell can be deconstructed into three distinct instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor the 16-cell.^{ [10] } The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.^{ [11] }

The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse golden ratio), but the 120-cell's edge length is ~0.270 times its radius.

Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius (φ^{2}/√8 ≈ 0.926) and edge length of exactly 1/4. Thus the unit-edge-length 120-cell (with long radius φ^{2}√2 ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4.

Reciprocally, the 120-cell whose coordinates are given above of long radius √8 = 2√2 ≈ 2.828 and edge length 2/φ^{2} = 3−√5 ≈ 0.764 can be constructed just outside a 600-cell of slightly smaller long radius, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 600-cell must have 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 600-cell, so that must be φ.

Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways).^{ [lower-alpha 3] } The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells.

The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.^{ [13] } 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 obviously).^{ [14] } This shows that the 120-cell contains, among its many interior features, 120 compounds of ten tetrahedra.

All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.^{ [lower-alpha 6] } Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells.

Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing 4-pyramids of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into several 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron.

Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.^{ [lower-alpha 7] } The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.

This configuration matrix represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{ [15] }^{ [16] }

Here is the configuration expanded with *k*-face elements and *k*-figures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.

H_{4} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | k-fig | 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 | 720 | 3 | 3 | {3} | H_{4}/A_{2}A_{1} = 14400/6/2 = 1200 | |

H_{2}A_{1} | {5} | f_{2} | 5 | 5 | 1200 | 2 | { } | H_{4}/H_{2}A_{1} = 14400/10/2 = 720 | |

H_{3} | {5,3} | f_{3} | 20 | 30 | 12 | 120 | ( ) | H_{4}/H_{3} = 14400/120 = 120 |

The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24-cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.^{ [17] }

The cell locations lend themselves to a hyperspherical description.^{ [18] } Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12).

Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles.

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 | Non-meridian / interstitial | 60° | |

4 | 12 cells | Second layer of meridional cells / "Tropic of Cancer" | 72° | |

5 | 30 cells | Non-meridian / interstitial | 90° | Equator |

6 | 12 cells | Third layer of meridional cells / "Tropic of Capricorn" | 108° | Southern Hemisphere |

7 | 20 cells | Non-meridian / interstitial | 120° | |

8 | 12 cells | Fourth layer of meridional cells / "Antarctic Circle" | 144° | |

9 | 1 cell | South Pole | 180° | |

Total | 120 cells |

The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell.

The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.^{ [19] }^{ [20] } Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical torsion. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two (Clifford) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.^{ [21] } Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint (Clifford parallel) great circles.

There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 cells and 6 edges. Both the above great circle paths have dual great circle paths in the 600-cell. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a decagon. The alternating cell/edge path above maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six triangular bipyramids) in the 600-cell. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub 24-cell (or icosahedral pyramids in the 600-cell).

Orthogonal projections of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by B. L. Chilton.^{ [22] }

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) |

3-dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image.

3D isometric projection | Animated 4D rotation |

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. Schlegel diagrams use perspective to show four-dimensional figures, choosing a point *above* a specific cell, thus making that cell the envelope of the 3D model, with other cells appearing smaller inside it. Stereographic projections use the same approach, but are shown with curved edges, representing the polytope as a tiling of a 3-sphere.

A comparison of perspective projections from 3D to 2D is shown in analogy.

Projection | Dodecahedron | 120-cell |
---|---|---|

Schlegel diagram | 12 pentagon faces in the plane | 120 dodecahedral cells in 3-space |

Stereographic projection | With transparent faces |

Perspective projection | |
---|---|

Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: - Nearest dodecahedron to the 4D viewpoint rendered in yellow
- The 12 dodecahedra immediately adjoining it rendered in cyan;
- The remaining dodecahedra rendered in green;
- Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image.
| |

Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: - Four cells surrounding nearest vertex shown in 4 colors
- Nearest vertex shown in white (center of image where 4 cells meet)
- Remaining cells shown in transparent green
- Cells facing away from 4D viewpoint culled for clarity
| |

A 3D projection of a 120-cell performing a simple rotation. | |

A 3D projection of a 120-cell performing a simple rotation (from the inside). | |

Animated 4D rotation |

The 120-cell is one of 15 regular and uniform polytopes with the same H_{4} symmetry [3,3,5]:^{ [23] }

H_{4} family polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

120-cell | rectified 120-cell | truncated 120-cell | cantellated 120-cell | runcinated 120-cell | cantitruncated 120-cell | runcitruncated 120-cell | omnitruncated 120-cell | ||||

{5,3,3} | r{5,3,3} | t{5,3,3} | rr{5,3,3} | t_{0,3}{5,3,3} | tr{5,3,3} | t_{0,1,3}{5,3,3} | t_{0,1,2,3}{5,3,3} | ||||

600-cell | rectified 600-cell | truncated 600-cell | cantellated 600-cell | bitruncated 600-cell | cantitruncated 600-cell | runcitruncated 600-cell | omnitruncated 600-cell | ||||

{3,3,5} | r{3,3,5} | t{3,3,5} | rr{3,3,5} | 2t{3,3,5} | tr{3,3,5} | t_{0,1,3}{3,3,5} | t_{0,1,2,3}{3,3,5} |

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

{p,3,3} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | S^{3} | H^{3} | |||||||||

Form | Finite | Paracompact | Noncompact | ||||||||

Name | {3,3,3} | {4,3,3} | {5,3,3} | {6,3,3} | {7,3,3} | {8,3,3} | ...{∞,3,3} | ||||

Image | |||||||||||

Cells {p,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |

The 120-cell is a part of a sequence of 4-polytopes and honeycombs with dodecahedral cells:

{5,3,p} polytopes | |||||||
---|---|---|---|---|---|---|---|

Space | S^{3} | H^{3} | |||||

Form | Finite | Compact | Paracompact | Noncompact | |||

Name | {5,3,3} | {5,3,4} | {5,3,5} | {5,3,6} | {5,3,7} | {5,3,8} | ... {5,3,∞} |

Image | |||||||

Vertex figure | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |

The **Davis 120-cell**, introduced by Davis (1985), is a compact 4-dimensional hyperbolic manifold obtained by identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4-dimensional hyperbolic space.

- Uniform 4-polytope family with [5,3,3] symmetry
- 57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra.
- 600-cell - the dual 4-polytope to the 120-cell

- ↑ In the 120-cell, 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] } - 1 2 The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is
*rounder*than its predecessor, enclosing more 4-content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell 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 120-cell is the 600-point 4-polytope: sixth and last in the ascending sequence that begins with the 5-point 4-polytope. - 1 2 The vertices of the 120-cell can be partitioned into those of five disjoint 600-cells in two different ways.
^{ [12] } - ↑ In the 120-cell, each 24-cell belongs to two different 600-cells.
^{ [8] }. The 120-cell can be partitioned into 25 disjoint 24-cells.^{ [9] } - ↑ In the dodecahedral cell of the unit-radius 120-cell, the dodecahedron (120-cell) edge length is 1/φ
^{2}√2 ≈ 0.270. The 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 1st chord of the 120-cell). 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 (600-cell edges, and the 2nd chord of the 120-cell). The diameter of the dodecahedron is √3/φ√2 ≈ 0.757 (the cube diagonal, and the 4th chord of the 120-cell). - ↑ The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the ten must belong to a different 600-cell.
- ↑ As we saw in the 600-cell, these 12 tetrahedra belong (in pairs) to the 6 icosahedral clusters of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.

- ↑ N.W. Johnson:
*Geometries and Transformations*, (2018) ISBN 978-1-107-10340-5 Chapter 11:*Finite Symmetry Groups*, 11.5*Spherical Coxeter groups*, p.249 - ↑ Matila Ghyka,
*The Geometry of Art and Life*(1977), p.68 - ↑ Coxeter 1973, pp. 292–293, Table I(ii); "120-cell".
- ↑ Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.
- ↑
*Mathematics and Its History*, John Stillwell, 1989, 3rd edition 2010, ISBN 0-387-95336-1 - ↑ Stillwell 2001.
- ↑ Coxeter 1973, p. 157, §8.7 Cartesian coordinates.
- ↑ van Ittersum 2020, p. 435, §4.3.5 The two 600-cells circumscribing a 24-cell.
- ↑ 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. 3–4, §2 Geometry of the 120-cell: rays and bases; "The 120-cell has 600 vertices distributed symmetrically on the surface of a [3-sphere] in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300
**rays**[or axes] of the 120-cell. 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." - ↑ Waegell & Aravind 2014, pp. 5–6.
- ↑ Sullivan 1991, pp. 4–5, 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 mirror-image) is said to be
*chiral*." - ↑ Coxeter 1973, §1.8 Configurations.
- ↑ Coxeter 1991, p. 117.
- ↑ Sullivan 1991, p. 15, Other Properties of the 120-cell.
- ↑ Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.
- ↑ Zamboj 2021, pp. 6–12, §2 Mathematical background.
- ↑ Schleimer & Segerman 2013, pp. 16–18, §6.2. Rings of dodecahedra.
- ↑ Zamboj 2021, pp. 23–29, §5 Hopf tori corresponding to circles on B
^{2}. - ↑ Chilton 1964.
- ↑ Denney et al. 2020.

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In geometry, an **icosidodecahedron** is a polyhedron with twenty (*icosi*) triangular faces and twelve (*dodeca*) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, the **24-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called **C _{24}**, or the

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In geometry, the **600-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the **C _{600}**,

In geometry, the **5-cell** is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a **C _{5}**,

In geometry, the **16-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In geometry, the **rectified 600-cell** or **rectified hexacosichoron** is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

In geometry, the **snub 24-cell** or **snub disicositetrachoron** is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

In geometry, the **icosahedral honeycomb** is one of four compact, regular, space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

In geometry, the **grand antiprism** or **pentagonal double antiprismoid** is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope, discovered in 1965 by Conway and Guy. Topologically, under its highest symmetry, the pentagonal antiprisms have *D _{5d}* symmetry and there are two types of tetrahedra, one with

In four-dimensional geometry, a **runcinated 120-cell** is a convex uniform 4-polytope, being a runcination of the regular 120-cell.

A **regular dodecahedron** or **pentagonal dodecahedron** is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

In mathematics, a **regular 4-polytope** is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

The **Boerdijk–Coxeter helix**, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.

In geometry, a **pentagonal polytope** is a regular polytope in *n* dimensions constructed from the H_{n} Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3^{n − 2}} (dodecahedral) or {3^{n − 2}, 5} (icosahedral).

In the field of hyperbolic geometry, the **order-5 hexagonal tiling honeycomb** arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.

- Coxeter, H.S.M. (1973) [1948].
*Regular Polytopes*(3rd ed.). New York: Dover. - Coxeter, H.S.M. (1991).
*Regular Complex Polytopes*(2nd ed.). Cambridge: Cambridge University Press. - Coxeter, H.S.M. (1995). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.).
*Kaleidoscopes: Selected Writings of H.S.M. Coxeter*(2nd ed.). Wiley-Interscience Publication. ISBN 978-0-471-01003-6.- (Paper 22) H.S.M. Coxeter,
*Regular and Semi Regular Polytopes I*, [Math. Zeit. 46 (1940) 380-407, MR 2,10] - (Paper 23) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559-591] - (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 22) H.S.M. Coxeter,
- Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938).
*The Fifty-Nine Icosahedra*. Vol. 6. University of Toronto Studies (Mathematical Series). - Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF).
*Notices of the AMS*.**48**(1): 17–25. - J.H. Conway and M.J.T. Guy:
*Four-Dimensional Archimedean Polytopes*, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965 - N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. Dissertation, University of Toronto, 1966 - Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation
- Davis, Michael W. (1985), "A hyperbolic 4-manifold",
*Proceedings of the American Mathematical Society*,**93**(2): 325–328, doi:10.2307/2044771, ISSN 0002-9939, MR 0770546 - Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes".
*Advances in Geometry*.**20**(3): 433–444. arXiv: 1912.06156v1 . - Steinbach, Peter (1997). "Golden fields: A case for the Heptagon".
*Mathematics Magazine*.**70**(Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048. - Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths". arXiv: 1903.06971 [math.MG].
- Miyazaki, Koji (1990). "Primary Hypergeodesic Polytopes".
*International Journal of Space Structures*.**5**(3–4): 309–323. doi:10.1177/026635119000500312. - van Ittersum, Clara (2020). "Symmetry groups of regular polytopes in three and four dimensions".
*TUDelft*. - Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes".
*Symmetry*.**2**: 1423–1449. doi:10.3390/sym2031423. - Sullivan, John M. (1991). "Generating and Rendering Four-Dimensional Polytopes".
*Mathematica Journal*.**1**(3): 76–85. - Waegell, Mordecai; Aravind, P.K. (10 Sep 2014). "Parity proofs of the Kochen-Specker theorem based on the 120-cell". arXiv: 1309.7530v3 [quant-ph].
- Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in the double orthogonal projection of the 4-space". arXiv: 2003.09236v2 [math.HO].
- Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope".
*W:European Physical Journal E*.**5**: 575–582. doi: 10.1007/s101890170040 . S2CID 121229939. - Chilton, B. L. (September 1964). "On the projection of the regular polytope {5,3,3} into a regular triacontagon".
*Canadian Mathematical Bulletin*.**7**(3): 385–398. doi: 10.4153/CMB-1964-037-9 . - Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell" (PDF).
*Notices Amer. Math. Soc*.**62**(11): 1309–1316.

- Weisstein, Eric W. "120-Cell".
*MathWorld*. - Olshevsky, George. "Hecatonicosachoron".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5x - hi".
- Der 120-Zeller (120-cell) Marco Möller's Regular polytopes in R
^{4}(German) - 120-cell explorer – A free interactive program that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL.
- Construction of the Hyper-Dodecahedron
- YouTube animation of the construction of the 120-cell Gian Marco Todesco.

H_{4} family polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

120-cell | rectified 120-cell | truncated 120-cell | cantellated 120-cell | runcinated 120-cell | cantitruncated 120-cell | runcitruncated 120-cell | omnitruncated 120-cell | ||||

{5,3,3} | r{5,3,3} | t{5,3,3} | rr{5,3,3} | t_{0,3}{5,3,3} | tr{5,3,3} | t_{0,1,3}{5,3,3} | t_{0,1,2,3}{5,3,3} | ||||

600-cell | rectified 600-cell | truncated 600-cell | cantellated 600-cell | bitruncated 600-cell | cantitruncated 600-cell | runcitruncated 600-cell | omnitruncated 600-cell | ||||

{3,3,5} | r{3,3,5} | t{3,3,5} | rr{3,3,5} | 2t{3,3,5} | tr{3,3,5} | t_{0,1,3}{3,3,5} | t_{0,1,2,3}{3,3,5} |

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