Snub disicositetrachoron

Snub disicositetrachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
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	$\frac{1+\sqrt5}{2} ≈ 1.61803$
Hypervolume	$\frac{45+20\sqrt5}{4} ≈ 22.43034$
Dichoral angles	Tet–3–tet: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751°$
	Ike–3–tet: $\arccos\left(-\frac{\sqrt{10}}{4}\right) ≈ 142.23876°$
	Ike–3–ike: 120°
Central density	1
Number of external pieces	144
Level of complexity	10
Related polytopes
Army	Sadi
Regiment	Sadi
Dual	Quatro-icositetradiminished hexacosichoron
Conjugate	Retrosnub disicositetrachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4/2, order 576
Convex	Yes
Nature	Tame

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.

Gallery

• 

  Wireframe

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  Net

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  Card with cell counts, verf, and cross-sections

Vertex coordinates

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

• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12,\,0\right).$

Representations

A snub disicositetrachoron has the following Coxeter diagrams:

• s3s4o3o (full symmetry)
• o4s3s3s (BC4/2 symmetry, as alternated great rhombated hexadecachoron)
• s3s3s *b3s (D4+ symmetry, snub demitesseract)
• fox3ooo3xfo *b3oxf&#zx (D4 subsymmetry)
• oooxxxfffFFF FxfoFfxFofxo xfFFfoFoxxof fFxfoFoxFofx &#zx (A1×A1×A1×A1 subsymmetry)

General variant

The snub disicositetrachoron has a general isogonal variant that maintains its full symmetry. This variant uses 24 pyritohedral icosahedra, 24 regular tetrahedra, and 96 triangular pyramids, with three edge lengths. In particular, the variant derived from alternating a uniform truncated icositetrachoron has tetrahedra of size $\sqrt2$, pyramids with that edge length for base and $\sqrt3$ for sides, and icosahedral variants with $\sqrt3$ for 24 edges and $\sqrt2$ 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.

o3o4o3o truncations

Name	OBSA	CD diagram	Picture
Icositetrachoron	ico	x3o4o3o ()
Truncated icositetrachoron	tico	x3x4o3o ()
Rectified icositetrachoron	rico	o3x4o3o ()
Tetracontoctachoron	cont	o3x4x3o ()
Rectified icositetrachoron	rico	o3o4x3o ()
Truncated icositetrachoron	tico	o3o4x3x ()
Icositetrachoron	ico	o3o4o3x ()
Small rhombated icositetrachoron	srico	x3o4x3o ()
Great rhombated icositetrachoron	grico	x3x4x3o ()
Small rhombated icositetrachoron	srico	o3x4o3x ()
Great rhombated icositetrachoron	grico	o3x4x3x ()
Small prismatotetracontoctachoron	spic	x3o4o3x ()
Prismatorhombated icositetrachoron	prico	x3x4o3x ()
Prismatorhombated icositetrachoron	prico	x3o4x3x ()
Great prismatotetracontoctachoron	gippic	x3x4x3x ()
Snub disicositetrachoron	sadi	s3s4o3o ()

o4o3o3o truncations

Name	OBSA	CD diagram	Picture
Tesseract	tes	x4o3o3o ()
Truncated tesseract	tat	x4x3o3o ()
Rectified tesseract	rit	o4x3o3o ()
Tesseractihexadecachoron	tah	o4x3x3o ()
Rectified hexadecachoron = Icositetrachoron	ico	o4o3x3o ()
Truncated hexadecachoron	thex	o4o3x3x ()
Hexadecachoron	hex	o4o3o3x ()
Small rhombated tesseract	srit	x4o3x3o ()
Great rhombated tesseract	grit	x4x3x3o ()
Small rhombated hexadecachoron = Rectified icositetrachoron	rico	o4x3o3x ()
Great rhombated hexadecachoron = Truncated icositetrachoron	tico	o4x3x3x ()
Small disprismatotesseractihexadecachoron	sidpith	x4o3o3x ()
Prismatorhombated hexadecachoron	proh	x4x3o3x ()
Prismatorhombated tesseract	prit	x4o3x3x ()
Great disprismatotesseractihexadecachoron	gidpith	x4x3x3x ()

External links

• Bowers, Jonathan. "Category 20: Miscellaneous" (#969).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "sadi".

• Quickfur. "The Snub 24-cell".

• Wikipedia Contributors. "Snub 24-cell".
• Hi.gher.Space Wiki Contributors. "Snub demitesseract".