In geometry, the 600cell is the convex regular 4polytope (fourdimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C_{600}, hexacosichoron^{[1]} and hexacosihedroid.^{[2]} It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
600cell  

Type  Convex regular 4polytope 
Schläfli symbol  {3,3,5} 
Coxeter diagram  
Cells  600 (3.3.3) 
Faces  1200 {3} 
Edges  720 
Vertices  120 
Vertex figure  icosahedron 
Petrie polygon  30gon 
Coxeter group  H_{4}, [3,3,5], order 14400 
Dual  120cell 
Properties  convex, isogonal, isotoxal, isohedral 
Uniform index  35 
The 600cell'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 4dimensional 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 120cell.
The 600cell is the fifth in the sequence of 6 convex regular 4polytopes (in order of size and complexity).^{[b]} It can be deconstructed into twentyfive overlapping instances of its immediate predecessor the 24cell,^{[4]} as the 24cell can be deconstructed into three overlapping instances of its predecessor the tesseract (8cell), and the 8cell can be deconstructed into two overlapping instances of its predecessor the 16cell.^{[5]}
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 24cell's edge length equals its radius, but the 600cell's edge length is ~0.618 times its radius. The 600cell's radius and edge length are in the golden ratio.
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
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 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  1 16cell  2 16cells  3 8cells  25 24cells  10 600cells  
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 30gons  20 30gons  
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  
4Content 
The vertices of a 600cell of unit radius centered at the origin of 4space, with edges of length 1/φ ≈ 0.618 (where φ = 1 + √5/2 ≈ 1.618 is the golden ratio), can be given^{[6]} as follows:
8 vertices obtained from
by permuting coordinates, and 16 vertices of the form:
The remaining 96 vertices are obtained by taking even permutations of
Note that the first 8 are the vertices of a 16cell, the second 16 are the vertices of a tesseract, and those 24 vertices together are the vertices of a 24cell. The remaining 96 vertices are the vertices of a snub 24cell, which can be found by partitioning each of the 96 edges of another 24cell (dual to the first) in the golden ratio in a consistent manner.^{[7]}
When interpreted as quaternions, these are the unit icosians.
In the 24cell, there are squares, hexagons and triangles that lie on great circles (in central planes through four or six vertices).^{[d]} In the 600cell there are twentyfive overlapping inscribed 24cells, with each square unique to one 24cell, each hexagon or triangle shared by two 24cells, and each vertex shared among five 24cells.^{[f]}
In the 600cell there are also great circle pentagons and decagons (in central planes through ten vertices).^{[j]}
Only the decagon edges are visible elements of the 600cell (because they are the edges of the 600cell). The edges of the other great circle polygons are interior chords of the 600cell, which are not shown in any of the 600cell 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 nonorthogonal hexagons between which all axis pairs are orthogonal, or as 144 nonorthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600cell is captured by the set of Hopf coordinates^{[o]} (𝜉_{i}, 𝜂, 𝜉_{j}) given as:
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 𝜉_{i} and 𝜉_{j} 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]}
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.^{[10]} These can be seen in the H3 Coxeter plane projections with overlapping vertices colored.^{[11]}^{[12]}
These polyhedral sections are solids in the sense that they are 3dimensional, but of course all of their vertices lie on the surface of the 600cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4dimensional space as a parallel cross section through the 600cell (a hyperplane). In the curved 3dimensional space of the 600cell'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 600cell, 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 600cell. Thus V is the apex of a 4pyramid based on the polyhedron.
Concentric Hulls  

The 600cell 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: 
The 120 vertices are distributed^{[13]} at eight different chord lengths from each other. These edges and chords of the 600cell are simply the edges and chords of its five great circle polygons.^{[14]} 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 24cell (√1, √2, √3, √4) alternate with the four new chords of the 600cell'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]}
The 600cell rounds out the 24cell by adding 96 more vertices between the 24cell's existing 24 vertices,^{[v]} in effect adding twentyfour more overlapping 24cells inscribed in the 600cell.^{[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 24cells, which become invisible interior chords in the 600cell, like the √2 and √3 chords.
Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of 1/φ, the inverse golden ratio), the 600cell does not have unit edgelength in a unitradius coordinate system the way the 24cell and the tesseract do; unlike those two, the 600cell 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 twentyfive envelopes of 24 octahedral cells (adding some 4dimensional space in places between these curved 3dimensional envelopes). The shape of those interstices must be an octahedral 4pyramid of some kind, but in the 600cell it is not regular.^{[z]}
The vertex chords of the 600cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.^{[16]}
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 600cell 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 600cell. 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 nonintersecting 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.^{[18]}
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 24cells, with each hexagon in two of the 24cells). Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24cells. The √1 chords join vertices which are two √0.𝚫 edges apart. Each √1 chord is the long diameter of a facebonded 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 nonorthogonal (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.^{[19]} The 200 great hexagons can be divided into 10 sets of 20 nonintersecting 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.^{[20]}
The √1.𝚫 chords form 144 central pentagons, 6 of which cross at each vertex.^{[j]} The √1.𝚫 chords run vertextoeverysecondvertex 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 24cells). Each set of 18 squares consists of the 72 √2 chords and 24 vertices of one of the 25 overlapping inscribed 24cells. 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 24cell. 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 nonintersecting 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.^{[21]}
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 vertextoeverythirdvertex 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 24cells, with each triangle in two of the 24cells). Each set of 32 triangles consists of the 96 √3 chords and 24 vertices of one of the 25 overlapping inscribed 24cells. The √3 chords run vertextoeverysecondvertex 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 24cell.^{[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 vertextoeveryfourthvertex 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 600cell. 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 24cells.^{[h]}
The sum of the squared lengths^{[af]} of all these distinct chords of the 600cell is 14,400 = 120^{2}.^{[ag]} These are all the central polygons through vertices, but the 600cell does have one noteworthy great circle that does not pass through any vertices.^{[ak]} Moreover, in 4space there are geodesics on the 3sphere which do not lie in central planes at all. There are geodesic shortest paths between two 600cell 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 24cells), and square planes (𝜋/2 apart, also in the 75 inscribed 16cells and the 24cells). These central planes of the 600cell can be divided into 4 central hyperplanes (3spaces) 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 30gon plane.^{[ai]}
Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of nonintersecting 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 600cell, visiting all 120 vertices just once.^{[25]} The great circle polygons in each bundle spiral around each other, delineating helical rings of facebonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.^{[26]}
The fibrations of the 600cell 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.^{[27]} 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]}
The fibrations of the 24cell 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 600cell contains 25 24cells, and can be seen (10 different ways) as a compound of 5 disjoint 24cells.^{[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 24cells of 4 helical rings each; each hexagonal fibration of the 600cell consists of 5 disjoint 24cells.
The fibrations of the 16cell 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 nonintersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration. The 600cell contains 75 16cells, and can be seen (10 different ways) as a compound of 15 disjoint 16cells. 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.
Because each 16cell constitutes an orthonormal basis for the choice of a coordinate reference frame, the fibrations of different 16cells have different natural reference frames. The 15 fibrations of great squares in the 600cell correspond to the 15 natural reference frames of the 600cell. One or more of these reference frames is natural to each fibration of the 600cell. Each fibration of great hexagons has three (equally natural) of these reference frames (as the 24cell has 3 16cells); each fibration of great decagons has all 15 (as the 600cell has 15 disjoint 16cells).
The densely packed helical cell rings of fibrations are celldisjoint, but they share vertices, edges and faces. Each fibration of the 600cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings facebonded 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 facebonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.^{[28]} 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 600cell). 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 24cell's great hexagons can be seen (three different ways) as 2 completely disjoint cell rings with spaces separating them, rather than as 4 facebonded 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 600cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.
The fibrations of the 16cell's great squares can be seen (two different ways) as a single cell ring with an adjacent cellringsized empty space, rather than as 2 facebonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600cell's great squares can be seen as a single tetrahedral cell ring.^{[ar]}
The sparse constructions of the 600cell's fibrations correspond to lowersymmetry decompositions of the 600cell, 24cell or 16cell with cells of different colors to distinguish the cell rings from the spaces between them.^{[au]} The particular lowersymmetry form of the 600cell 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 2sphere). Each of the 20 BoerdijkCoxeter cell rings^{[ab]} is lifted from a corresponding face of the icosahedron.^{[aw]}
The 600cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5cell, the 120cell, and the polygons {7} and above.^{[29]} Consequently, there are numerous ways to construct or deconstruct the 600cell, but none of them are trivial. The construction of the 600cell from its regular predecessor the 24cell can be difficult to visualize.
Thorold Gosset discovered the semiregular 4polytopes, including the snub 24cell with 96 vertices, which falls between the 24cell and the 600cell in the sequence of convex 4polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600cell from the 24cell is in two steps, using the snub 24cell as an intermediate form. In the first, more complex step (described elsewhere) the snub 24cell is constructed by a special snub truncation of a 24cell at the golden sections of its edges.^{[7]} In the second step the 600cell is constructed in a straightforward manner by adding 4pyramids (vertices) to facets of the snub 24cell.^{[30]}
The snub 24cell is a diminished 600cell 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 24cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.
Constructing the unitradius 600cell from its precursor the unitradius 24cell by Gosset's method actually requires three steps. The 24cell precursor to the snub24 cell is not of the same radius: it is larger, since the snub24 cell is its truncation. Starting with the unitradius 24cell, the first step is to reciprocate it around its midsphere to construct its outer canonical dual: a larger 24cell, since the 24cell is selfdual. That larger 24cell can then be snub truncated into a unitradius snub 24cell.
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 3dimensional surface envelope,^{[x]} or how they lie on the underlying surface envelope of the 24cell's octahedral cells. For that it is helpful to build up the 600cell directly from clusters of tetrahedral cells.
Most of us have difficulty visualizing the 600cell from the outside in 4space, or recognizing an outside view of the 600cell due to our total lack of sensory experience in 4dimensional spaces,^{[31]} but we should be able to visualize the surface envelope of 600 cells from the inside because that volume is a 3dimensional space that we could actually "walk around in" and explore.^{[32]} In these exercises of building the 600cell up from cell clusters, we are entirely within a 3dimensional 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.
The vertex figure of the 600cell 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 600cell has a dihedral angle of 𝜋/3 + arccos(−1/4) ≈ 164.4775°.^{[34]}
An entire 600cell can be assembled from 24 such icosahedral pyramids (bonded facetoface at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells facebonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is facebonded 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 24cells inscribed in the 600cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed snub 24cell, which has exactly the same structure of icosahedra and tetrahedra described here, except that the icosahedra are not 4pyramids filled by tetrahedral cells; they are only "flat" 3dimensional icosahedral cells.
The 24cell 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 24cells,^{[bb]} and the 120 vertices comprise 25 (not 5) 24cells.^{[w]}
The icosahedra are facebonded 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 facetoface and vertextovertex. The long diameter of each facebonded pair of tetrahedra (each triangular bipyramid) is a hexagon edge (a 24cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apexvertex, just as there are 4 celldisjoint interlocking rings of 6 octahedra in the 24cell (a hexagonal fibration).^{[bd]}
The tetrahedral cells are facebonded 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 600cell.
The partitioning of the 600cell 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 600cell.^{[ay]} Their 120 apexes are each a vertex of five 24vertex 24cells, so there are 5*120/24 = 25 overlapping 24cells.^{[j]}
There is another useful way to partition the 600cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure^{[38]} and a direct construction of the 600cell from its predecessor the 24cell.
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 600cell 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 facebonded tetrahedra (triangular dipyramids) whose long diameter is a 24cell 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 24cell edges of length √1. They form a tetrahedron of edge length √1, which is the second section of the 600cell beginning with a cell.^{[bf]} There are 600 of these √1 tetrahedral sections in the 600cell.^{[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 600cell beginning with a cell, an octahedron of edge length √1, obviously the cell of a 24cell. 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 onesixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.^{[bi]}
Thus the unitradius 600cell may be constructed directly from its predecessor,^{[z]} the unitradius 24cell, 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.
There is yet another useful way to partition the 600cell surface into clusters of tetrahedral cells, which reveals more structure^{[39]} and the decagonal fibrations of the 600cell. An entire 600cell can be assembled from 2 rings of 5 icosahedral pyramids, bonded vertextovertex into geodesic "straight lines", plus 40 10cell rings which fill the voids remaining between the icosahedra.
The 120cell can be decomposed into two disjoint tori.^{[40]} Since it is the dual of the 600cell, this same dual tori structure exists in the 600cell, although it is somewhat more complex. The 10cell geodesic path in the 120cell corresponds to the 10vertex decagon path in the 600cell.
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 triangularfaced column with a pentagonal crosssection.^{[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 10vertex 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 tetrahedraloctahedral 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 600cell has symmetry [[10,2^{+},10]], order 400, the same symmetry as the grand antiprism. The grand antiprism is just the 600cell with the two above 150cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous to the 10face belt of an icosahedron with the 5 top and 5 bottom faces removed (a pentagonal antiprism).^{[bs]}
The two 150cell 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 600cell with cells.
A single 30tetrahedron Boerdijk–Coxeter helix ring within the 600cell, seen in stereographic projection.^{[ab]} 
A 30tetrahedron ring can be seen along the perimeter of this 30gonal orthogonal projection.^{[ak]} 
The 30 vertices of the 30cell ring lie on a skew star 30gon with a winding number of 11.^{[aj]} 
The 600cell can also be partitioned into 20 celldisjoint intertwining rings of 30 cells, each ten edges long, forming a discrete Hopf fibration which fills the entire 600cell.^{[41]} These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix.^{[ab]} The center axis of each helix is a great 30gon geodesic that does not intersect any vertices.^{[ak]} The 30 vertices of the 30cell ring form a skew compound 30gon with a geodesic orbit that winds around the 600cell twice.^{[aj]} The dual of the 30cell ring (the 30gon made by connecting its cell centers) is a skew 30gon Petrie polygon.^{[bw]} Five of these 30cell helices nest together and spiral around each of the 10vertex decagon paths, forming the 150cell torus described above.^{[42]} Thus every great decagon is the center core decagon of a 150cell torus.^{[bx]}
The 20 celldisjoint 30cell rings constitute four identical celldisjoint 150cell tori: the two described in the grand antiprism decomposition above, and two more that fill the middle layer of 300 tetrahedra occupied by 30 10cell rings in the grand antiprism decomposition.^{[by]} The four 150cell rings spiral around each other and pass through each other in the same manner as the 20 30cell rings or the 12 great decagons; these three sets of Clifford parallel polytopes are the same discrete decagonal fibration of the 600cell.^{[br]}
The regular convex 4polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations about a fixed point in 4dimensional Euclidean space.^{[cd]}
The 600cell is generated by isoclinic rotations^{[al]} of the 24cell by 36° = 𝜋/5 (the arc of one 600cell edge length).^{[cf]}
There are 25 inscribed 24cells in the 600cell. Therefore there are also 25 inscribed snub 24cells, 75 inscribed tesseracts and 75 inscribed 16cells.^{[w]}
The 8vertex 16cell 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 24vertex 24cell 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 16cells, isoclinically rotated by 𝜋/3 with respect to each other.
The 120vertex 600cell has 60 long diameters: not just 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24cells.^{[44]} There are 5 disjoint 24cells in the 600cell, but not just 5: there are 10 different ways to partition the 600cell into 5 disjoint 24cells.^{[h]}
Like the 16cells and 8cells inscribed in the 24cell, the 25 24cells inscribed in the 600cell are mutually isoclinic polytopes. The rotational distance between inscribed 24cells is always an equalangled rotation of 𝜋/5 in each pair of completely orthogonal invariant planes of rotation.^{[ce]}
Five 24cells are disjoint because they are Clifford parallel: their corresponding vertices are 𝜋/5 apart on two nonintersecting 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 24cell to a disjoint 24cell (just as an isoclinic rotation of hexagonal planes by 𝜋/3 takes each 16cell to a disjoint 16cell).^{[cg]} Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24cells to the left of each 24cell, and another 4 disjoint 24cells to its right.^{[ci]} The left and right rotations reach different 24cells; therefore each 24cell belongs to two different sets of five disjoint 24cells.
All Clifford parallel polytopes are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).^{[cj]} Each 24cell 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]} Nondisjoint 24cells are related by a simple rotation by 𝜋/5 in an invariant plane intersecting only two vertices of the 600cell, 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 24cell to another 24cell.^{[cg]} Disjoint 24cells 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]} Nondisjoint 24cells 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 24cells is Clifford parallel because its corresponding great hexagons are Clifford parallel. (24cells do not have great decagons.) The 16 great hexagons in each 24cell can be divided into 4 sets of 4 nonintersecting Clifford parallel geodesics, each set of which covers all 24 vertices of the 24cell. The 200 great hexagons in the 600cell can be divided into 10 sets of 20 nonintersecting 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 24cell. Similarly, the corresponding great squares of disjoint 24cells are Clifford parallel.
The 600cell can be constructed radially from 720 golden triangles of edge lengths √0.𝚫 √1 √1 which meet at the center of the 4polytope, 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 600cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5cells with regular √0.𝚫 tetrahedron bases (the cells of the 600cell).
This configuration matrix^{[45]} represents the 600cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Here is the configuration expanded with kface elements and kfigures. 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}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  kfig  Notes  

H_{3}  ( )  f_{0}  120  12  30  20  {3,5}  H_{4}/H_{3} = 14400/120 = 120  
A_{1}H_{2}  { }  f_{1}  2  720  5  5  {5}  H_{4}/H_{2}A_{1} = 14400/10/2 = 720  
A_{2}A_{1}  {3}  f_{2}  3  3  1200  2  { }  H_{4}/A_{2}A_{1} = 14400/6/2 = 1200  
A_{3}  {3,3}  f_{3}  4  6  4  600  ( )  H_{4}/A_{3} = 14400/24 = 600 
The icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600cell.^{[46]} The icosians lie in the golden field, (a + b√5) + (c + d√5)i + (e + f√5)j + (g + h√5)k, where the eight variables are rational numbers.^{[47]} The finite sums of the 120 unit icosians are called the icosian ring.
When interpreted as quaternions, the 120 vertices of the 600cell 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 600cell as an invariant subgroup, namely as the subgroup 2I_{L} of quaternion leftmultiplications and as the subgroup 2I_{R} of quaternion rightmultiplications. Each rotational symmetry of the 600cell is generated by specific elements of 2I_{L} and 2I_{R}; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the nonrotation Id and the central inversion −Id. We have the isomorphism RSG ≅ (2I_{L} × 2I_{R}) / {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 600cell is the Weyl group of H_{4}.^{[48]} This is a group of order 14400. It consists of 7200 rotations and 7200 rotationreflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss.^{[49]}
The symmetries of the 3D surface of the 600cell 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 600cell is the dual of the 120cell. One may also notice that the 600cell also contains the vertices of a dodecahedron,^{[29]} which with some effort can be seen in most of the below perspective projections.
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=1,orange=5,yellow=10) 
[6] (Red=1,orange=3,yellow=6) 
[4] (Red=1,orange=2,yellow=4) 
A threedimensional model of the 600cell, 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.^{[50]}
Vertexfirst projection  

This image shows a vertexfirst perspective projection of the 600cell into 3D. The 600cell is scaled to a vertexcenter radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
 
Cellfirst projection  
This image shows the 600cell in cellfirst perspective projection into 3D. Again, the 600cell to a vertexcenter radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
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 

The snub 24cell may be obtained from the 600cell by removing the vertices of an inscribed 24cell and taking the convex hull of the remaining vertices. This process is a diminishing of the 600cell.
The grand antiprism may be obtained by another diminishing of the 600cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.
A bi24diminished 600cell, with all tridiminished icosahedron cells has 48 vertices removed, leaving 72 of 120 vertices of the 600cell. The dual of a bi24diminished 600cell, is a tri24diminished 600cell, with 48 vertices and 72 hexahedron cells.
There are a total of 314,248,344 diminishings of the 600cell by nonadjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.^{[51]}
Diminished 600cells  

Name  Tri24diminished 600cell  Bi24diminished 600cell  Snub 24cell (24diminished 600cell) 
Grand antiprism (20diminished 600cell) 
600cell  
Vertices  48  72  96  100  120  
Vertex figure (Symmetry) 
dual of tridiminished icosahedron ([3], order 6) 
tetragonal antiwedge ([2]^{+}, order 2) 
tridiminished icosahedron ([3], order 6) 
bidiminished icosahedron ([2], 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 H_{4} plane 

Ortho F_{4} plane 
The regular complex polytopes _{3}{5}_{3}, and _{5}{3}_{5}, , in have a real representation as 600cell in 4dimensional space. Both have 120 vertices, and 120 edges. The first has Complex reflection group _{3}[5]_{3}, order 360, and the second has symmetry _{5}[3]_{5}, order 600.^{[52]}
Regular complex polytope in orthogonal projection of H_{4} Coxeter plane  

{3,3,5} Order 14400 
_{3}{5}_{3} Order 360 
_{5}{3}_{5} Order 600 
The 600cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:
H_{4} family polytopes  

120cell  rectified 120cell 
truncated 120cell 
cantellated 120cell 
runcinated 120cell 
cantitruncated 120cell 
runcitruncated 120cell 
omnitruncated 120cell  
{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}  
600cell  rectified 600cell 
truncated 600cell 
cantellated 600cell 
bitruncated 600cell 
cantitruncated 600cell 
runcitruncated 600cell 
omnitruncated 600cell  
{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} 
It is similar to three regular 4polytopes: the 5cell {3,3,3}, 16cell {3,3,4} of Euclidean 4space, and the order6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.
{3,3,p} polytopes  

Space  S^{3}  H^{3}  
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 4polytope is a part of a sequence of 4polytope and honeycombs with icosahedron vertex figures:
{p,3,5} polytopes  

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