Snub 24-cell

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Snub 24-cell
(No image)
Type	Uniform polychoron
Cells	96 3.3.3 (oblique) 24 3.3.3 24 3.3.3.3.3
Faces	480 {3}
Edges	432
Vertices	96
Vertex figure	5 3.3.3 3 3.3.3.3.3 (Tridiminished icosahedron)
Schläfli symbol	s{3,4,3}
Coxeter-Dynkin diagram
Symmetry group	[3+,4,3]
Properties	convex

Vertex figure: Tridiminished icosahedron
8 faces:

5 3.3.3 and 3 3.3.3.3.3

In geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedra and 24 icosahedra cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

It is one of three semiregular polychora made of two or more types of cells which are platonic solids.

It is related to the truncated 24-cell by an alternated truncation. That is, half the vertices are deleted, the 24 truncated octahedron cells become 24 icosahedron cells, the 24 cubes become 24 tetrahedron cells, and the 96 deleted vertex voids create 96 new tetrahedron cells.

Names:

Snub icositetrachoron
Snub 24-cell
Snub polyoctahedron
Sadi (Jonathan Bowers: for snub disicositetrachoron)

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+√5)/2 is the golden ratio).

These 96 vertices can be found by partitioning each of the 96 edges of a 24-cell into the golden ratio in a consistent manner, in much the same way that 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 is 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. The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of the 600-cell.