BREAKING NEWS

## Summary

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.

600-cell Schlegel diagram, vertex-centered
(vertices and edges)
TypeConvex regular 4-polytope
Schläfli symbol{3,3,5}
Coxeter diagram       Cells600 (3.3.3) Faces1200 {3}
Edges720
Vertices120
Vertex figure icosahedron
Petrie polygon30-gon
Coxeter groupH4, [3,3,5], order 14400
Dual120-cell
Propertiesconvex, isogonal, isotoxal, isohedral
Uniform index35 The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex.[a] Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell.

## Geometry

The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[b] It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell, as the 24-cell can be deconstructed into three overlapping instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two overlapping instances of its predecessor the 16-cell.

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.[c] The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius. The 600-cell's radius and edge length are in the golden ratio.

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 dihedrals 𝝅/2 𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2 𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2 𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/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 1 16-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 $1$  $1$  $1$  $1$  $1$  $1$
Edge length ${\sqrt {\tfrac {5}{2}}}\approx 1.581$  ${\sqrt {2}}\approx 1.414$  $1$  $1$  ${\tfrac {1}{\phi }}\approx 0.618$  ${\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270$
Short radius ${\tfrac {1}{4}}$  ${\tfrac {1}{2}}$  ${\tfrac {1}{2}}$  ${\sqrt {\tfrac {1}{2}}}\approx 0.707$  $1-\left({\tfrac {\sqrt {2}}{2\phi {\sqrt {3}}}}\right)^{2}\approx 0.936$  $1-\left({\tfrac {1}{2\phi {\sqrt {3}}}}\right)^{2}\approx 0.968$
Area $10\left({\sqrt {\tfrac {8}{9}}}\right)\approx 9.428$  $32\left({\sqrt {\tfrac {3}{16}}}\right)\approx 13.856$  $24$  $96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$  $1200\left({\tfrac {\sqrt {3}}{8\phi ^{2}}}\right)\approx 99.238$  $720\left({\tfrac {25+10{\sqrt {5}}}{8\phi ^{4}}}\right)\approx 621.9$
Volume $5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$  $16\left({\tfrac {1}{3}}\right)\approx 5.333$  $8$  $24\left({\sqrt {\tfrac {2}{9}}}\right)\approx 11.314$  $600\left({\tfrac {1}{3\phi ^{3}{\sqrt {8}}}}\right)\approx 16.693$  $120\left({\tfrac {2+\phi }{2\phi ^{3}{\sqrt {8}}}}\right)\approx 18.118$
4-Content ${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$  ${\tfrac {2}{3}}\approx 0.667$  $1$  $2$  ${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.907$  ${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.385$

### Coordinates

The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length 1/φ ≈ 0.618 (where φ = 1 + 5/2 ≈ 1.618 is the golden ratio), can be given as follows:

8 vertices obtained from

(0, 0, 0, ±1)

by permuting coordinates, and 16 vertices of the form:

1/2, ±1/2, ±1/2, ±1/2)

The remaining 96 vertices are obtained by taking even permutations of

φ/2, ±1/2, ±φ−1/2, 0)

Note that the first 8 are the vertices of a 16-cell, the second 16 are the vertices of a tesseract, and those 24 vertices together are the vertices of a 24-cell. The remaining 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.

When interpreted as quaternions, these are the unit icosians.

In the 24-cell, there are squares, hexagons and triangles that lie on great circles (in central planes through four or six vertices).[d] In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each square unique to one 24-cell, each hexagon or triangle shared by two 24-cells, and each vertex shared among five 24-cells.[f]

#### Hopf spherical coordinates

In the 600-cell there are also great circle pentagons and decagons (in central planes through ten vertices).[j]

Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article.

By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of completely orthogonal[l] squares which do not share any vertices, or as 100 dual pairs of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of Hopf coordinates[o] (𝜉i, 𝜂, 𝜉j) given as:

({<10}𝜋/5, {≤5}𝜋/10, {<10}𝜋/5)

where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉i and 𝜉j coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.[p]

### Structure

#### Polyhedral sections

The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = 𝜋/5, 60° = 𝜋/3, 72° = 2𝜋/5, 90° = 𝜋/2, 108° = 3𝜋/5, 120° = 2𝜋/3, 144° = 4𝜋/5, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron,[a] at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V. These can be seen in the H3 Coxeter plane projections with overlapping vertices colored.

These polyhedral sections are solids in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a 4-pyramid based on the polyhedron.

Concentric Hulls
The 600-cell is projected to 3D using an orthonormal basis.

The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:

1) two points at the origin
2) two icosahedra
3) two dodecahedra
4) two larger icosahedra
5) and a single icosidodecahedron

for a total of 120 vertices. This is the view from any origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices.

#### Vertex chords

Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths[d] with angles of arc. The golden ratio[q] governs the fractional roots of every other chord,[r] and the radial golden triangles[s] which meet at the center.

The 120 vertices are distributed at eight different chord lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length, they are 0.𝚫, 1, 1.𝚫, 2, 2.𝚽, 3, 3.𝚽, and 4.[t]

Notice that the four hypercubic chords of the 24-cell (1, 2, 3, 4) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new chord lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio[q] including the two golden sections of 5, as shown in the diagram.[r]

#### Boundary envelopes

A 3D projection of a 600-cell performing a simple rotation. The 3D surface made of 600 tetrahedra is visible.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,[v] in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.[w] The new surface thus formed is a tessellation of smaller, more numerous cells[x] and faces: tetrahedra of edge length 1/φ ≈ 0.618 instead of octahedra of edge length 1. It encloses the 1 edges of the 24-cells, which become invisible interior chords in the 600-cell, like the 2 and 3 chords.

A 3D projection of a 24-cell performing a simple rotation. The 3D surface made of 24 octahedra is visible. It is also present in the 600-cell, but as an invisible interior boundary envelope.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of 1/φ, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not radially equilateral. Like them it is radially triangular in a special way, but one in which golden triangles rather than equilateral triangles meet at the center.[s]

The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an octahedral 4-pyramid of some kind, but in the 600-cell it is not regular.[z]

#### Geodesics

The vertex chords of the 600-cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.

Cell-centered stereographic projection of the 600-cell's 72 central decagons onto their great circles. Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.

The 0.𝚫 = 𝚽 edges form 72 flat regular central decagons, 6 of which cross at each vertex.[a] Just as the icosidodecahedron can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 0.𝚫 edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, 3.𝚽 apart. As in the decagon and the icosidodecahedron, the edges occur in golden triangles[y] which meet at the center of the polytope.[s] The 72 great decagons can be divided into 6 sets of 12 non-intersecting Clifford parallel geodesics,[i] such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.

The 1 chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),[e] 10 of which cross at each vertex[ad] (4 from each of five 24-cells, with each hexagon in two of the 24-cells). Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The 1 chords join vertices which are two 0.𝚫 edges apart. Each 1 chord is the long diameter of a face-bonded pair of tetrahedral cells (a triangular bipyramid), and passes through the center of the shared face. As there are 1200 faces, there are 1200 1 chords, in 600 parallel pairs, 3 apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 dual pairs in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual. The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.

The 1.𝚫 chords form 144 central pentagons, 6 of which cross at each vertex.[j] The 1.𝚫 chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. The 1.𝚫 chords join vertices which are two 0.𝚫 edges apart on a geodesic great circle. The 720 1.𝚫 chords occur in 360 parallel pairs, 2.𝚽 = φ apart.

The 2 chords form 450 central squares (25 disjoint sets of 18), 15 of which cross at each vertex (3 from each of five 24-cells). Each set of 18 squares consists of the 72 2 chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The 2 chords join vertices which are three 0.𝚫 edges apart (and two 1 chords apart). Each 2 chord is the long diameter of an octahedral cell in just one 24-cell. There are 1800 2 chords, in 900 parallel pairs, 2 apart. The 450 great squares (225 completely orthogonal[l] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares in each set reach all 120 vertices.

The 2.𝚽 = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length 3.𝚽. The 2.𝚽 chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three 0.𝚫 edges apart on a geodesic great circle. There are 720 distinct 2.𝚽 chords, in 360 parallel pairs, 1.𝚫 apart.

The 3 chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five 24-cells, with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 3 chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The 3 chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The 3 chords join vertices which are four 0.𝚫 edges apart (and two 1 chords apart on a geodesic great circle). Each 3 chord is the long diameter of two cubic cells in the same 24-cell.[ae] There are 1200 3 chords, in 600 parallel pairs, 1 apart.

The 3.𝚽 chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length 1.𝚫, so these are golden triangles.[y] The 3.𝚽 chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four 0.𝚫 edges apart on a geodesic great circle. There are 720 distinct 3.𝚽 chords, in 360 parallel pairs, 0.𝚫 apart.

The 4 chords occur as 60 long diameters (75 sets of 4 orthogonal axes), the 120 long radii of the 600-cell. The 4 chords join opposite vertices which are five 0.𝚫 edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.[h]

The sum of the squared lengths[af] of all these distinct chords of the 600-cell is 14,400 = 1202.[ag] These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices.[ak] Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they corresponding to isoclinic (diagonal) rotations rather than simple rotations.[al]

All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes (𝜋/5 apart), hexagon planes (𝜋/3 apart, also in the 25 inscribed 24-cells), and square planes (𝜋/2 apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 central hyperplanes (3-spaces) each forming an icosidodecahedron. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.[aq] Each great square plane is completely orthogonal[l] to another great square plane. Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one 4 long diameter): a great digon plane. Each great decagon plane is completely orthogonal to a plane which intersects no vertices: a great 30-gon plane.[ai]

#### Fibrations

Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).[i] Each fiber bundle of Clifford parallel great circles[am] is a discrete Hopf fibration which fills the 600-cell, visiting all 120 vertices just once. The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.

##### Decagons

The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.[ac] Each fiber bundle[an] delineates 20 helical rings of 30 tetrahedral cells each,[ab] with five rings nesting together around each decagon. Each tetrahedral cell occupies only one cell ring in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.[aa]

##### Hexagons

The fibrations of the 24-cell include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration. The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.[j] It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. Each fiber bundle[ao] delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 10 fibrations. The 20 helical rings belong to 5 disjoint 24-cells of 4 helical rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells.

##### Squares

The fibrations of the 16-cell include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration. The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. Each fiber bundle[ap] delineates 150 helical rings of 8 tetrahedral cells each.[ar] Each tetrahedral cell occupies only one cell ring in each of the 15 fibrations.

##### Reference frames

Because each 16-cell constitutes an orthonormal basis for the choice of a coordinate reference frame, the fibrations of different 16-cells have different natural reference frames. The 15 fibrations of great squares in the 600-cell correspond to the 15 natural reference frames of the 600-cell. One or more of these reference frames is natural to each fibration of the 600-cell. Each fibration of great hexagons has three (equally natural) of these reference frames (as the 24-cell has 3 16-cells); each fibration of great decagons has all 15 (as the 600-cell has 15 disjoint 16-cells).

##### Clifford parallel cell rings

The densely packed helical cell rings of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other. The same fibration can also be seen as a minimal sparse arrangement of fewer completely disjoint cell rings that do not touch at all.[as]

The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint tetrahedral cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon. The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices. This subset of 4 of 20 cell rings is dimensionally analogous to the subset of 12 of 72 decagons, in that both are sets of completely disjoint Clifford parallel polytopes which visit all 120 vertices.[at] The subset of 4 of 20 cell rings is one of 5 fibrations within the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with subfibrations.

The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.

The fibrations of the 16-cell's great squares can be seen (two different ways) as a single cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.[ar]

The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or 16-cell with cells of different colors to distinguish the cell rings from the spaces between them.[au] The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous to the snub tetrahedron form of the icosahedron (which is the base[av] of these fibrations on the 2-sphere). Each of the 20 Boerdijk-Coxeter cell rings[ab] is lifted from a corresponding face of the icosahedron.[aw]

### Constructions

The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above. Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize.

#### Gosset's construction

Thorold Gosset discovered the semiregular 4-polytopes, including the snub 24-cell with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described elsewhere) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the golden sections of its edges. In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.

The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,[v] leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.[a] The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.

Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires three steps. The 24-cell precursor to the snub-24 cell is not of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its midsphere to construct its outer canonical dual: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell.

#### Cell clusters

Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional surface envelope,[x] or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.

Most of us have difficulty visualizing the 600-cell from the outside in 4-space, or recognizing an outside view of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces, but we should be able to visualize the surface envelope of 600 cells from the inside because that volume is a 3-dimensional space that we could actually "walk around in" and explore. In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, closed curved space, in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.

##### Icosahedra

A regular icosahedron colored in snub octahedron symmetry.[ax] Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the icosahedral pyramid (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).

A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible). The four cells lie in different hyperplanes.

The vertex figure of the 600-cell is the icosahedron.[a] Twenty tetrahedral cells meet at each vertex, forming an icosahedral pyramid whose apex is the vertex, surrounded by its base icosahedron. The 600-cell has a dihedral angle of 𝜋/3 + arccos(−1/4) ≈ 164.4775°.

An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.[az].

The apexes of the 24 icosahedral pyramids are the vertices of 24-cells inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed snub 24-cell, which has exactly the same structure of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells.

The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.[ba] Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,[bb] and the 120 vertices comprise 25 (not 5) 24-cells.[w]

The icosahedra are face-bonded into geodesic "straight lines" by their opposite faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a great circle hexagon. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each triangular bipyramid) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking rings of 6 octahedra in the 24-cell (a hexagonal fibration).[bd]

The tetrahedral cells are face-bonded into triple helices, bent in the fourth dimension into rings of 30 tetrahedral cells.[ab] The three helixes are geodesic "straight lines" of 10 edges: great circle decagons which run Clifford parallel[i] to each other. Each tetrahedron, having six edges, participates in six different decagons[aa] and thereby in all 6 of the decagonal fibrations of the 600-cell.

The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.[ay] Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.[j]

##### Octahedra

There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure and a direct construction of the 600-cell from its predecessor the 24-cell.

Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell.

First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra (triangular dipyramids) whose long diameter is a 24-cell edge (a hexagon edge) of length 1. Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,[be] so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length 1. They form a tetrahedron of edge length 1, which is the second section of the 600-cell beginning with a cell.[bf] There are 600 of these 1 tetrahedral sections in the 600-cell.[bg]

With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length 1, obviously the cell of a 24-cell. As partially filled so far (by 17 tetrahedral cells), this 1 octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.[bh] Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.[bi]

Thus the unit-radius 600-cell may be constructed directly from its predecessor,[z] the unit-radius 24-cell, by placing on each of its octahedral facets a truncated[bj] irregular octahedral pyramid of 14 vertices[bk] constructed (in the above manner) from 25 regular tetrahedral cells of edge length 1/φ ≈ 0.618.

##### Union of two tori

There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure and the decagonal fibrations of the 600-cell. An entire 600-cell can be assembled from 2 rings of 5 icosahedral pyramids, bonded vertex-to-vertex into geodesic "straight lines", plus 40 10-cell rings which fill the voids remaining between the icosahedra.

100 tetrahedra in a 10×10 array forming a Clifford torus boundary in the 600 cell. Its opposite edges are identified, forming a duocylinder.[bm]

The 120-cell can be decomposed into two disjoint tori. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.

Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex stacked icosahedral pyramids, with the five extra annular ring gaps also filled in.[bn] The surface is the same as that of ten stacked pentagonal antiprisms: a triangular-faced column with a pentagonal cross-section.[bo] Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces, 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face.[bq] This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges. The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon, but mathematically they are all equivalent (they all lie in central planes).

Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a Clifford torus. They can be "unrolled" into a square 10x10 array.[bl] Incidentally this structure forms one tetrahedral layer in the tetrahedral-octahedral honeycomb. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori. In this case into each recess, instead of an octahedron as in the honeycomb, fits a triangular bipyramid composed of two tetrahedra.[br]

This decomposition of the 600-cell has symmetry [[10,2+,10]], order 400, the same symmetry as the grand antiprism. The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a pentagonal antiprism).[bs]

The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete fibration of 12 decagons that reaches all 120 vertices, despite filling only half the 600-cell with cells. A single 30-tetrahedron Boerdijk–Coxeter helix ring within the 600-cell, seen in stereographic projection.[ab] A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection.[ak] The 30 vertices of the 30-cell ring lie on a skew star 30-gon with a winding number of 11.[aj]

The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells, each ten edges long, forming a discrete Hopf fibration which fills the entire 600-cell. These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix.[ab] The center axis of each helix is a great 30-gon geodesic that does not intersect any vertices.[ak] The 30 vertices of the 30-cell ring form a skew compound 30-gon with a geodesic orbit that winds around the 600-cell twice.[aj] The dual of the 30-cell ring (the 30-gon made by connecting its cell centers) is a skew 30-gon Petrie polygon.[bw] Five of these 30-cell helices nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described above. Thus every great decagon is the center core decagon of a 150-cell torus.[bx]

The 20 cell-disjoint 30-cell rings constitute four identical cell-disjoint 150-cell tori: the two described in the grand antiprism decomposition above, and two more that fill the middle layer of 300 tetrahedra occupied by 30 10-cell rings in the grand antiprism decomposition.[by] The four 150-cell rings spiral around each other and pass through each other in the same manner as the 20 30-cell rings or the 12 great decagons; these three sets of Clifford parallel polytopes are the same discrete decagonal fibration of the 600-cell.[br]

#### Rotations

The regular convex 4-polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations about a fixed point in 4-dimensional Euclidean space.[cd]

The 600-cell is generated by isoclinic rotations[al] of the 24-cell by 36° = 𝜋/5 (the arc of one 600-cell edge length).[cf]

There are 25 inscribed 24-cells in the 600-cell. Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.[w]

The 8-vertex 16-cell has 4 long diameters inclined at 90° = 𝜋/2 to each other, often taken as the 4 orthogonal axes or basis of the coordinate system.

The 24-vertex 24-cell has 12 long diameters inclined at 60° = 𝜋/3 to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by 𝜋/3 with respect to each other.

The 120-vertex 600-cell has 60 long diameters: not just 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells. There are 5 disjoint 24-cells in the 600-cell, but not just 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.[h]

Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually isoclinic polytopes. The rotational distance between inscribed 24-cells is always an equal-angled rotation of 𝜋/5 in each pair of completely orthogonal invariant planes of rotation.[ce]

A 4-dimensional ring of three Clifford parallel great decagons, cut and laid out flat in 3 dimensional space.[ac]

Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are 𝜋/5 apart on two non-intersecting Clifford parallel[i] decagonal great circles (as well as 𝜋/5 apart on the same decagonal great circle).[ac] An isoclinic rotation of decagonal planes by 𝜋/5 takes each 24-cell to a disjoint 24-cell (just as an isoclinic rotation of hexagonal planes by 𝜋/3 takes each 16-cell to a disjoint 16-cell).[cg] Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the left of each 24-cell, and another 4 disjoint 24-cells to its right.[ci] The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells.

All Clifford parallel polytopes are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).[cj] Each 24-cell is isoclinic and Clifford parallel to 8 others, and isoclinic but not Clifford parallel to 16 others.[e] With each of the 16 it shares 6 vertices: a hexagonal central plane.[bb] Non-disjoint 24-cells are related by a simple rotation by 𝜋/5 in an invariant plane intersecting only two vertices of the 600-cell, a rotation in which the completely orthogonal fixed plane is their common hexagonal central plane. They are also related by an isoclinic rotation in which both planes rotate by 𝜋/5.[cl]

There are two kinds of 𝜋/5 isoclinic rotations which take each 24-cell to another 24-cell.[cg] Disjoint 24-cells are related by a 𝜋/5 isoclinic rotation of an entire fibration of 12 Clifford parallel decagonal invariant planes. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.)[ci] Non-disjoint 24-cells are related by a 𝜋/5 isoclinic rotation of an entire fibration of 20 Clifford parallel hexagonal invariant planes.[cn] (There are 10 such sets of fibers, so there are 20 such distinct rotations.)[ck]

On the other hand, each of the 10 sets of five disjoint 24-cells is Clifford parallel because its corresponding great hexagons are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel geodesics, each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, each set of which covers all 120 vertices and constitutes a discrete hexagonal fibration. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great squares of disjoint 24-cells are Clifford parallel.

The 600-cell can be constructed radially from 720 golden triangles of edge lengths 0.𝚫 1 1 which meet at the center of the 4-polytope, each contributing two 1 radii and a 0.𝚫 edge.[s] They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral 0.𝚫 bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular 0.𝚫 tetrahedron bases (the cells of the 600-cell).

### As a configuration

This configuration matrix represents the 600-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 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

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

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 fk f0 f1 f2 f3 k-fig Notes
H3         ( ) f0 120 12 30 20 {3,5} H4/H3 = 14400/120 = 120
A1H2         { } f1 2 720 5 5 {5} H4/H2A1 = 14400/10/2 = 720
A2A1         {3} f2 3 3 1200 2 { } H4/A2A1 = 14400/6/2 = 1200
A3         {3,3} f3 4 6 4 600 ( ) H4/A3 = 14400/24 = 600

## Symmetries

The icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The icosians lie in the golden field, (a + b5) + (c + d5)i + (e + f5)j + (g + h5)k, where the eight variables are rational numbers. The finite sums of the 120 unit icosians are called the icosian ring.

When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2IL of quaternion left-multiplications and as the subgroup 2IR of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion −Id. We have the isomorphism RSG ≅ (2IL × 2IR) / {Id, -Id}. The order of RSG equals 120 × 120/2 = 7200.

The binary icosahedral group is isomorphic to SL(2,5).

The full symmetry group of the 600-cell is the Weyl group of H4. This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss.

## Visualization

The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,[x] and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron, which with some effort can be seen in most of the below perspective projections.

### 2D projections

The H3 decagonal projection shows the plane of the van Oss polygon.

Orthographic projections by Coxeter planes
H4 - F4


(Red=1)


(Red=1)


(Red=1)
H3 A2 / B3 / D4 A3 / B2


(Red=1,orange=5,yellow=10)


(Red=1,orange=3,yellow=6)


(Red=1,orange=2,yellow=4)

### 3D projections

A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.

Vertex-first projection
This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
• The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown.
• The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow.
• The remaining cells are rendered in edge-outline.
• Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image.
Cell-first projection
This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
• The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image.
• The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow.
• The remaining cells are rendered in edge-outline.
• Cells facing away from the 4D viewpoint have been culled for clarity.

This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.

Frame synchronized orthogonal isometric (left) and perspective (right) projections

## Diminished 600-cells

The snub 24-cell may be obtained from the 600-cell by removing the vertices of an inscribed 24-cell and taking the convex hull of the remaining vertices. This process is a diminishing of the 600-cell.

The grand antiprism may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.

A bi-24-diminished 600-cell, with all tridiminished icosahedron cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells.

There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.

Diminished 600-cells
Name Tri-24-diminished 600-cell Bi-24-diminished 600-cell Snub 24-cell
(24-diminished 600-cell)
Grand antiprism
(20-diminished 600-cell)
600-cell
Vertices 48 72 96 100 120
Vertex figure
(Symmetry)

dual of tridiminished icosahedron
(, order 6)

tetragonal antiwedge
(+, order 2)

tridiminished icosahedron
(, order 6)

bidiminished icosahedron
(, order 4)

Icosahedron
([5,3], order 120)
Symmetry Order 144 (48×3 or 72×2) [3+,4,3]
Order 576 (96×6)
[[10,2+,10]]
Order 400 (100×4)
[5,3,3]
Order 14400 (120×120)
Net
Ortho
H4 plane

Ortho
F4 plane

## Related complex polygons

The regular complex polytopes 3{5}3,     and 5{3}5,    , in $\mathbb {C} ^{2}$  have a real representation as 600-cell in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has Complex reflection group 33, order 360, and the second has symmetry 55, order 600.

Regular complex polytope in orthogonal projection of H4 Coxeter plane

{3,3,5}
Order 14400

3{5}3
Order 360

5{3}5
Order 600

## Related polytopes and honeycombs

The 600-cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:

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}

It is similar to three regular 4-polytopes: the 5-cell {3,3,3}, 16-cell {3,3,4} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.

{3,3,p} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3}

{3,3,4}

{3,3,5}

{3,3,6}

{3,3,7}

{3,3,8}

... {3,3,∞}

Image
Vertex
figure

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,∞}

This 4-polytope is a part of a sequence of 4-polytope and honeycombs with icosahedron vertex figures:

{p,3,5} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {3,3,5}

{4,3,5}

{5,3,5}

{6,3,5}

{7,3,5}

{8,3,5}

... {∞,3,5}

Image
Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{∞,3}