Snub disicositetrachoron
Snub disicositetrachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Sadi 
Coxeter diagram  s3s4o3o () 
Elements  
Cells  24+96 tetrahedra, 24 icosahedra 
Faces  96+96+288 triangles 
Edges  144+288 
Vertices  96 
Vertex figure  Tridiminished icosahedron, edge length 1 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Tet–3–tet: 
Ike–3–tet:  
Ike–3–ike: 120°  
Central density  1 
Number of external pieces  144 
Level of complexity  10 
Related polytopes  
Army  Sadi 
Regiment  Sadi 
Dual  Quatroicositetradiminished hexacosichoron 
Conjugate  Retrosnub disicositetrachoron 
Abstract & topological properties  
Flag count  5760 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  F_{4}/2, order 576 
Convex  Yes 
Nature  Tame 
The snub disicositetrachoron, or sadi, also commonly called the snub 24cell or snub demitesseract, is a convex uniform polychoron that consists of 24+96 regular tetrahedra and 24 regular icosahedra. 5 tetrahedra and 3 icosahedra join at each vertex, forming a tridiminished icosahedron as the vertex figure.
It can be constructed by alternating the vertices of a truncated icositetrachoron and then adjusting for equal edge lengths. Alternatively, it can be thought of as a diminishing of the regular hexacosichoron, where 24 vertices corresponding to the vertices of an inscribed icositetrachoron are removed. This is why an alternate valid Bowers style acronym would be idex.
Diminishing by a further set of an inscribed icositetrachoron's vertices of the snub disicositetrachoron yields a biicositetradiminished hexacosichoron, diminishing two such sets yields a triicositetradiminished hexacosichoron, diminishing three such sets yields a quatroicositetradiminished hexacosichoron, and diminishing all four of those here remaining subsets results in the hecatonicosachoron. It turns out that all of these pairs from zero to five diminishings result in pairs of dual polychora. In particular, the snub disicositetrachoron's dual is the quatroicositetradiminished hexacosichoron.
It belongs to a larger family of diminished hexacosichora known as the special cuts, which are a subset of the Blind polytopes.
Gallery[edit  edit source]

Wireframe

Net

Card with cell counts, verf, and crosssections
Vertex coordinates[edit  edit source]
The vertices of a snub disicositetrachoron of edge length 1, centered at the origin, are given by all even permutations of:
 .
Representations[edit  edit source]
A snub disicositetrachoron has the following Coxeter diagrams:
 s3s4o3o () (full symmetry)
 o4s3s3s () (B_{4}/2 symmetry, as alternated great rhombated hexadecachoron)
 s3s3s *b3s () (D_{4}+ symmetry, snub demitesseract)
 fox3ooo3xfo *b3oxf&#zx (D_{4} subsymmetry)
 oooxxxfffFFF FxfoFfxFofxo xfFFfoFoxxof fFxfoFoxFofx &#zx (K_{4} subsymmetry)
General variant[edit  edit source]
The snub disicositetrachoron has a general isogonal variant with two degrees of variation that maintains its full symmetry. This variant uses 24 pyritohedral icosahedra, 24 regular tetrahedra, and 96 triangular pyramids, with two edge lengths. In particular, the variant derived from alternating a uniform truncated icositetrachoron has tetrahedra of size , pyramids with that edge length for base and for sides, and icosahedral variants with for 24 edges and for the other 6.
Variations[edit  edit source]
The snub disicositetrachoron has two isogonal variants with lower symmetry:
 Snub rhombatohexadecachoron  Pyritotesseractic symmetry, has 8 pyritohedral icosahedra, 16 snub tetrahedra, 24 tetragonal disphenoids, and 96 sphenoids
 Snub demitesseract  Chiral demitesseract symmetry, has 3 sets of 8 snub tetrahedra, 24 rhombic disphenoids, and 96 irregular tetrahedra
Related polychora[edit  edit source]
The snub disicositetrachoron's regiment also contains a coincidic scaliform polychoron, the icositetradiminished faceted hexacosichoron.
External links[edit  edit source]
 Bowers, Jonathan. "Category 20: Miscellaneous" (#969).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "sadi".
 Quickfur. "The Snub 24cell".
 Wikipedia contributors. "Snub 24cell".
 Hi.gher.Space Wiki Contributors. "Snub demitesseract".