Snub 24-cell
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| Snub 24-cell | |
|---|---|
 Orthogonal projection Centered on hyperplane of one icosahedron. |
|
| 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{31,1,1} h0,1{3,4,3} h0,1,2{3,3,4} |
| Coxeter-Dynkin diagrams |
   or     or |
| Coxeter group | [3+,4,3] |
| Properties | convex |
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.
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[edit] Geometry
It is one of three semiregular polychora, discovered by Thorold Gosset, made from two types of cells which are platonic solids. (The others two are the rectified 5-cell and rectified 600-cell.)
It is related to the truncated 24-cell by an alternation operation. 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.
[edit] Coordinates
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.
[edit] Projections
Stereographic projection of snub 24-cell:
 Wireframe |
 Tetrahedra colored transparent green |
[edit] Symmetry
It has three vertex-transitive colorings based on a Wythoff construction on a Coxeter group from which it is alternated from: F4 defines 24 interchangable icosahedra, while the C4 group defines two groups of icosahedra in a 8:16 counts, and finally the B4 group has 3 groups of icosahedra with 8:8:8 counts.
- B4, [31,1,1]: Snub 24-cell: s{31,1,1} - Three icosahedral sets {8,8,8}
- Vertex figure:
(red, green, and blue icosahedra; yellow and cyan tetrahedra)
- Vertex figure:
- C4, [3,3,4]: Alternated cantitruncated 16-cell: h0,1,2{3,3,4} - Two icosahedral sets {8,16}
- Vertex figure:
(red and blue icosahedra; yellow and cyan tetrahedra)
- Vertex figure:
- F4, [3,4,3]: Alternated truncated 24-cell: h0,1{3,4,3} - One icosahedral set {24}
- Vertex figure:
(blue icosahedra; yellow and cyan tetrahedra)
- Vertex figure:
[edit] Alternative names
- Snub icositetrachoron
- Snub 24-cell
- Snub polyoctahedron
- Sadi (Jonathan Bowers: for snub disicositetrachoron)
- Tetricosahedric Thorold Gosset, 1900
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[edit] References
- H.S.M. Coxeter, Regular Polytopes, Dover Publications Inc., 1973, New York, pp. 151–153.
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
