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Concept
Hexacosichore
Résumé
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.
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.
The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).
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.
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 + /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.
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