# Snub disicositetrachoron

Snub disicositetrachoron
Rank4
TypeUniform
Notation
Coxeter diagrams3s4o3o ()
Elements
Cells24+96 tetrahedra, 24 icosahedra
Faces96+96+288 triangles
Edges144+288
Vertices96
Vertex figureTridiminished icosahedron, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Hypervolume${\displaystyle {\frac {45+20{\sqrt {5}}}{4}}\approx 22.43034}$
Dichoral anglesTet–3–tet: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Ike–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }}$
Ike–3–ike: 120°
Central density1
Number of external pieces144
Level of complexity10
Related polytopes
ConjugateRetrosnub disicositetrachoron
Abstract & topological properties
Flag count5760
Euler characteristic0
OrientableYes
Properties
SymmetryF4/2, order 576
ConvexYes
NatureTame

The snub disicositetrachoron, or sadi, also commonly called the snub 24-cell 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 bi-icositetradiminished hexacosichoron, diminishing two such sets yields a tri-icositetradiminished hexacosichoron, diminishing three such sets yields a quatro-icositetradiminished 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 quatro-icositetradiminished hexacosichoron.

It belongs to a larger family of diminished hexacosichora known as the special cuts, which are a subset of the Blind polytopes.

## Vertex coordinates

The vertices of a snub disicositetrachoron of edge length 1, centered at the origin, are given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right)}$.

## Representations

A snub disicositetrachoron has the following Coxeter diagrams:

• s3s4o3o () (full symmetry)
• o4s3s3s () (B4/2 symmetry, as alternated great rhombated hexadecachoron)
• s3s3s *b3s () (D4+ symmetry, snub demitesseract)
• fox3ooo3xfo *b3oxf&#zx (D4 subsymmetry)
• oooxxxfffFFF FxfoFfxFofxo xfFFfoFoxxof fFxfoFoxFofx &#zx (K4 subsymmetry)

## General variant

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 ${\displaystyle {\sqrt {2}}}$, pyramids with that edge length for base and ${\displaystyle {\sqrt {3}}}$ for sides, and icosahedral variants with ${\displaystyle {\sqrt {3}}}$ for 24 edges and ${\displaystyle {\sqrt {2}}}$ for the other 6.

## Variations

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

The snub disicositetrachoron's regiment also contains a coincidic scaliform polychoron, the icositetradiminished faceted hexacosichoron.