600-cell

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, with which it can form a compound.

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,[4] 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.[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 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.

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[6] as follows:

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.[7]

In the 24-cell, there are squares, hexagons and triangles that lie on great circles (in equatorial 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]

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

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,[h] hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 4 sets of 6 orthogonal central pentagons which do not share any vertices.[i] This pentagonal symmetry of the 600-cell is revealed by its Hopf coordinates[l] (𝜉i, 𝜂, 𝜉j) which can be given as:

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 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 that center of the polyhedron 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.

The 120 vertices are distributed[13] 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.[14] In ascending order of length, they are 0.𝚫, 1, 1.𝚫, 2, 2.𝚽, 3, 3.𝚽, and 4.[q]

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[n] including the two golden sections of 5, as shown in the diagram.[o]

The vertex chords of the 600-cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.[15]

The 0.𝚫 = 𝚽 edges form 72 flat regular central decagons (12 sets of 6 orthogonal planes), 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[s] which meet at the center of the polytope.[p]

The 1 chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets), 10 of which cross at each vertex[t] (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 1.𝚫 chords form 144 central pentagons, 6 of which cross at each vertex.[g] 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 edges 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 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.[u] 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.[s] 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.[v]

The sum of the squared lengths[w] of all these distinct chords of the 600-cell is 14,400 = 1202.[x] These are all the geodesics through vertices, but the 600-cell does have at least one other geodesic that does not pass through any vertices.[y]

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.[z] The new surface thus formed is a tessellation of smaller, more numerous cells[aa] 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 interior chords in the 600-cell, 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 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.[p]

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 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.[ab]

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.[7] 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.[19]

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, 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.

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 3-dimensional surface envelope,[aa] 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.[20] In this exercise of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, closed curved one, in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.

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°.[22]

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. 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 the icosahedral pyramid, and three from a cluster of 5 cells (one of which is the central tetrahedron of the five). 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).

The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell 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 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. Thus there are 120 overlapping icosahedra in the 600-cell.

Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.[ad] Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells, and the 120 vertices comprise 25 (not 5) 24-cells.[z]

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).

The tetrahedral cells are face-bonded into helices, bent in the fourth dimension into rings of 30 tetrahedral cells.[ae] Their edges form geodesic "straight lines" of 10 edges: great circle decagons. Each tetrahedron, having six edges, participates in six different decagons.

There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure[23] 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,[af] 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.[ag] There are 600 of these 1 tetrahedral sections in the 600-cell.[ah]

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.[ai] Each octahedral cell consists of 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.

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

The 600-cell is generated by rotations of the 24-cell in increments of 36° = 𝜋/5 (the arc of one 600-cell edge length).

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.[z]

The 8-vertex 16-cell has 4 long diameters inclined at 90° = 𝜋/2 to each other, often taken as the 4 orthogonal axes 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.[v]

The 24-cells are rotated with respect to each other in increments of 𝜋/5. The rotational distance between inscribed 24-cells is always a double rotation of 0 to 4 increments of 𝜋/5 in one invariant plane, combined with 0 to 4 increments of 𝜋/5 in the orthogonal invariant plane, where the invariant planes are any two orthogonal central decagons. The product of these two 5-click simple rotations produces 25 distinct ways we can pick the 24 vertices of a 24-cell out of the 120 vertices of a 600-cell.

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.[p] 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).

This configuration matrix[24] 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.

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.

The icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell.[25] 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.[26] The finite sums of the 120 unit icosians are called the icosian ring.

The full symmetry group of the 600-cell is the Weyl group of H4.[27] 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.[28]

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,[aa] 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,[18] which with some effort can be seen in most of the below perspective projections.

100 tetrahedra in a 10×10 array forming a clifford torus boundary in the 600 cell.

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 a 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 "saucer" with 10 tetrahedra forming an icosahedron. You can view this as five, vertex stacked, icosahedral pyramids, with the five extra annular ring gaps also filled in. The surface is the same as that of ten stacked pentagonal antiprisms. You now have 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. 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. These paths spiral around the center core path, but mathematically they are all equivalent. 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. 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.

The 600-cell can be further partitioned into 20 disjoint intertwining rings of 30 cells and ten edges long each, forming a discrete Hopf fibration.[29] These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix. Five such helices nest and spiral around each of the 10-vertex decagon paths, forming the initial 150 cell torus mentioned above. The center axis of each helix is a 30-gon geodesic that does not intersect any vertices.[y]

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 tetrahedra, similar to the belt of an icosahedron with the 5 top and 5 bottom triangles removed (pentagonal antiprism).

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.[30]

Frame synchronized animated comparison of the 600 cell using orthogonal isometric (left) and perspective (right) projections.

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.[31]

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

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