120-cell

Last updated
120-cell
Schlegel diagram
(vertices and edges)
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 H4, [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 C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron [1] and hecatonicosahedroid. [2]

Contents

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.

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, [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 A4 B4 F4 H4
Name 5-cell

Hyper-tetrahedron
5-point

16-cell

Hyper-octahedron
8-point

8-cell

Hyper-cube
16-point

24-cell

24-point

600-cell

Hyper-icosahedron
120-point

120-cell

Hyper-dodecahedron
600-point

Schläfli symbol {3, 3, 3}{3, 3, 4}{4, 3, 3}{3, 4, 3}{3, 3, 5}{5, 3, 3}
Coxeter mirrors
Mirror dihedrals𝝅/2𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2
Graph
Vertices581624120600
Edges102432967201200
Faces10 triangles32 triangles24 squares96 triangles1200 triangles720 pentagons
Cells5 tetrahedra16 tetrahedra8 cubes24 octahedra600 tetrahedra120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed120 in 120-cell675 in 120-cell2 16-cells3 8-cells25 24-cells10 600-cells
Great polygons 2 𝝅/2 squares x 34 𝝅/2 rectangles x 34 𝝅/3 hexagons x 412 𝝅/5 decagons x 650 𝝅/15 dodecagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Isocline polygons 1 2 2 4 20
Long radius${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$
Edge length${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$${\displaystyle {\sqrt {2}}\approx 1.414}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius${\displaystyle {\tfrac {1}{4}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area${\displaystyle 10\left({\sqrt {\tfrac {8}{9}}}\right)\approx 9.428}$${\displaystyle 32\left({\sqrt {\tfrac {3}{16}}}\right)\approx 13.856}$${\displaystyle 24}$${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$${\displaystyle 1200\left({\tfrac {\sqrt {3}}{8\phi ^{2}}}\right)\approx 99.238}$${\displaystyle 720\left({\tfrac {25+10{\sqrt {5}}}{8\phi ^{4}}}\right)\approx 621.9}$
Volume${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$${\displaystyle 8}$${\displaystyle 24\left({\sqrt {\tfrac {2}{9}}}\right)\approx 11.314}$${\displaystyle 600\left({\tfrac {1}{3\phi ^{3}{\sqrt {8}}}}\right)\approx 16.693}$${\displaystyle 120\left({\tfrac {2+\phi }{2\phi ^{3}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$${\displaystyle {\tfrac {2}{3}}\approx 0.667}$${\displaystyle 1}$${\displaystyle 2}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

Cartesian coordinates

The 600 vertices of a 120-cell with an edge length of 2/φ2 = 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, ±φ2)

and all even permutations of

(0, ±φ−2, ±1, ±φ2)
(0, ±φ−1, ±φ, ±5)
(±φ−1, ±1, ±φ, ±2)

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

Polyhedral graph

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 42 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/φ2, 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.

Constructions

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.

Dual 600-cells

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 (φ2/8 ≈ 0.926) and edge length of exactly 1/4. Thus the unit-edge-length 120-cell (with long radius φ22 ≈ 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 = 22 ≈ 2.828 and edge length 2/φ2 = 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 φ2, 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.

Cell rotations of inscribed duals

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.

Augmentation

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.

As a configuration

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]

${\displaystyle {\begin{bmatrix}{\begin{matrix}600&4&6&4\\2&1200&3&3\\5&5&720&2\\20&30&12&120\end{matrix}}\end{bmatrix}}}$

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.

H4 k-face fkf0f1f2f3 k-fig Notes
A3( )f0600464 {3,3} H4/A3 = 14400/24 = 600
A1A2{ }f1272033 {3} H4/A2A1 = 14400/6/2 = 1200
H2A1 {5} f25512002{ }H4/H2A1 = 14400/10/2 = 720
H3 {5,3} f3203012120( )H4/H3 = 14400/120 = 120

Visualization

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]

Layered stereographic projection

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 CellsDescriptionColatitudeRegion
11 cellNorth PoleNorthern Hemisphere
212 cellsFirst layer of meridional cells / "Arctic Circle"36°
320 cellsNon-meridian / interstitial60°
412 cellsSecond layer of meridional cells / "Tropic of Cancer"72°
530 cellsNon-meridian / interstitial90°Equator
612 cellsThird layer of meridional cells / "Tropic of Capricorn"108°Southern Hemisphere
720 cellsNon-meridian / interstitial120°
812 cellsFourth layer of meridional cells / "Antarctic Circle"144°
91 cellSouth Pole180°
Total120 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 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.

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 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).

Projections

Orthogonal projections

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.

Orthographic projections by Coxeter planes
H4-F4

[30]
(Red=1)

[20]
(Red=1)

[12]
(Red=1)
H3A2 / B3 / D4A3 / B2

[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

Perspective projections

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.

Comparison with regular dodecahedron
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

H4 polytopes

The 120-cell is one of 15 regular and uniform polytopes with the same H4 symmetry [3,3,5]: [23]

H4 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}t0,3{5,3,3}tr{5,3,3}t0,1,3{5,3,3}t0,1,2,3{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}t0,1,3{3,3,5}t0,1,2,3{3,3,5}

{p,3,3} polytopes

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 S3 H3
FormFiniteParacompactNoncompact
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}

{5,3,p} polytopes

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

{5,3,p} polytopes
SpaceS3 H3
FormFiniteCompactParacompactNoncompact
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,}

Davis 120-cell

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.

Notes

1. 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]
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.
3. The vertices of the 120-cell can be partitioned into those of five disjoint 600-cells in two different ways. [12]
4. 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]
5. In the dodecahedral cell of the unit-radius 120-cell, the dodecahedron (120-cell) edge length is 1/φ22 ≈ 0.270. The orange vertices lie at the Cartesian coordinates (±φ38, ±φ38, ±φ38) relative to origin at the cell center. They form a cube (dashed lines) of edge length 1/φ2 ≈ 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/φ2 ≈ 0.757 (the cube diagonal, and the 4th chord of the 120-cell).
6. 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.
7. 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.

Citations

1. 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
2. Matila Ghyka, The Geometry of Art and Life (1977), p.68
3. Coxeter 1973, pp. 292–293, Table I(ii); "120-cell".
4. Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}.
5. Mathematics and Its History, John Stillwell, 1989, 3rd edition 2010, ISBN   0-387-95336-1
6. Coxeter 1973, p. 157, §8.7 Cartesian coordinates.
7. van Ittersum 2020, p. 435, §4.3.5 The two 600-cells circumscribing a 24-cell.
8. Denney et al. 2020, p. 5, §2 The Labeling of H4.
9. Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
10. 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."
11. Waegell & Aravind 2014, pp. 5–6.
12. Sullivan 1991, pp. 4–5, The Dodecahedron.
13. 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 ."
14. Coxeter 1973, §1.8 Configurations.
15. Coxeter 1991, p. 117.
16. Sullivan 1991, p. 15, Other Properties of the 120-cell.
17. Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra.
18. Zamboj 2021, pp. 6–12, §2 Mathematical background.
19. Schleimer & Segerman 2013, pp. 16–18, §6.2. Rings of dodecahedra.
20. Zamboj 2021, pp. 23–29, §5 Hopf tori corresponding to circles on B2.

Related Research Articles

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 C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

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 C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.

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 C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

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 C16, hexadecachoron, or hexdecahedroid [sic?].

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 D5d symmetry and there are two types of tetrahedra, one with S4 symmetry and one with Cs symmetry.

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 Hn 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, 3n − 2} (dodecahedral) or {3n − 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.

References

H4 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}t0,3{5,3,3}tr{5,3,3}t0,1,3{5,3,3}t0,1,2,3{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}t0,1,3{3,3,5}t0,1,2,3{3,3,5}
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds