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Concept
Snub 24-cell
Summary
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.
Topologically, under its highest symmetry, [3+,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with Th symmetry), 24 regular tetrahedra, and 96 triangular pyramids.
It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetricosahedric for being made of tetrahedron and icosahedron cells. (The other two are the rectified 5-cell and rectified 600-cell.)
Snub icositetrachoron
Snub demitesseract
Semi-snub polyoctahedron (John Conway)
Sadi (Jonathan Bowers) for snub disicositetrachoron
Tetricosahedric (Thorold Gosset)
The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking even permutations of
(0, ±1, ±φ, ±φ2)
where φ = 1+/2 ≈ 1.618 is the golden ratio.
The unit-radius coordinates of the snub 24-cell, with edges of length φ−1 ≈ 0.618, are the even permutations of
(±φ/2, ±1/2, ±φ−1/2, 0)
These 96 vertices can be found by partitioning each of the 96 edges of a 24-cell in the golden ratio in a consistent manner dimensionally analogous to the way the 12 vertices of an icosahedron or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This can be done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. This is equivalent to the snub truncation construction of the 24-cell described below.
Official source
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