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,N.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249 hexacosihedroidMatila Ghyka, The Geometry of Art and Life (1977), p.68 and hypericosahedron. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. 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 complexity(as measured by comparing configuration matrices or simply the number of vertices) and enclosed volume at the same radius. 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 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. The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius, which is the golden ratio.
Coordinates
Unit radius Cartesian coordinates The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length (where is the golden ratio), can be given 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 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). In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells. In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells. Each 16-cell constitutes a distinct orthonormal basis for the choice of a coordinate reference frame.
The 60 axes and 75 16-cells of the 600-cell constitute a geometric configuration, which in the language of configurations is written as 605754 to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes. Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs.
Hopf spherical coordinates In the 600-cell there are also great circle pentagons and decagons (in central planes through ten vertices).
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 (except where shown as dashed lines).
The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a snub 24-cell) by the 600-cell's 144 pentagons. Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex. Five 24-cells meet at each 600-cell vertex, so all 25 24-cells are linked by each great pentagon. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways), and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 decagonal fibrations) by choosing a pentagon from the same fibration at each 24-cell vertex.
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 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 (𝜉i, 𝜂, 𝜉j) given as:
({<10}, {≤5}, {<10})
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.
Structure
Polyhedral sections The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = , 60° = , 72° = , 90° = , 108° = , 120° = , 144° = , and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, 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 three-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 surface 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.
The vertex first envelope of the {3,3,5} 600-cell forms the pentakis icosidodecahedron (aka. the Class I geodesic polyhedron), which is the dual of the chamfered dodecahedron (the cell first envelope of the {5,3,3} 120-cell). It contains 42 vertices with 30 from the icosidodecahedron and 12 from the icosahedron shown below.
Concentric HullsThe 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.
Golden chords
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:
In the diagram, chord lengths are given as square roots, with a decimal fractional part if necessary, where:
is the inverse golden ratio, and:
is its square. For example, the 600-cell edge length is:
The four hypercubic chords of the 24-cell (, , , ) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new golden chord lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio including the two golden sections of .
Boundary envelopes
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. The new surface thus formed is a tessellation of smaller, more numerous cells and faces: tetrahedra of edge length instead of octahedra of edge length . It encloses the edges of the 24-cells, which become invisible interior chords in the 600-cell, like the and chords.
Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of , 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. Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) Radially golden polytopes are those which can be constructed, with their radii, from golden triangles.
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.
Geodesics The vertex chords of the 600-cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.
The edges form 72 flat regular central decagons, 6 of which cross at each vertex. 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 edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, apart. As in the decagon and the icosidodecahedron, the edges occur in golden triangles which meet at the center of the polytope. The 72 great decagons can be divided into 6 sets of 12 non-intersecting Clifford parallel geodesics, 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 chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets) 10 of which cross at each vertex (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 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 chords join vertices which are two edges apart. Each 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 chords, in 600 parallel pairs, apart.