Small ditetrahedronary hexacosihecatonicosachoron

Small ditetrahedronary hexacosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sidtaxhi
Coxeter diagram
Elements
Cells	600 tetrahedra, 120 small ditrigonary icosidodecahedra
Faces	2400 triangles, 720 pentagrams
Edges	3600
Vertices	600
Vertex figure	Semi-uniform Truncated tetrahedron, edge lengths 1 (triangle edges) and (√5–1)/2 (other edges)
Edge figure	(tet 3 sidtid 5/2 sidtid 3)
Measures (edge length 1)
Circumradius	$\frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825$
Hypervolume	$5\frac{100+37\sqrt5}{4} ≈ 228.41814$
Dichoral angles	Sidtid–3–tet: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
	Sidtid–5/2–sidtid: 144°
Central density	2
Number of pieces	2520
Level of complexity	9
Related polytopes
Army	Hi
Regiment	Sidtaxhi
Conjugate	Grand ditetrahedronary hexacosihecatonicosachoron
Abstract properties
Euler characteristic	—600
Topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small ditetrahedronary hexacosihecatonicosachoron, or sidtaxhi, is a nonconvex uniform polychoron that consists of 600 regular tetrahedra and 120 small ditrigonary icosidodecahedra. 4 small ditrigonary icosidodecahedra and 4 tetrahedra join at each vertex, with a variant of the truncated tetrahedron as the vertex figure.

The small ditetrahedronary hexacosihecatonicosachoron contains the vertices of a small rhombicosidodecahedral prism and decagonal duoprism.

It can be formed as a holosnub hecatonicosachoron.

Vertex coordinates

The vertices of a small ditetrahedronary hexacosihecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,0,\,0\right),$
• $\left(±\frac{5+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{\sqrt5-1}{4}\right),$
• $\left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12,\,±\frac12\right),$

together with all the even permutations of:

• $\left(±\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,0\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0,\,±\frac12\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac12\right).$

Related polychora

The small ditetrahedronary hexacosihecatonicosachoron is the colonel of a regiment with 37 members, plus 5 fissaries and three compounds (two are subsymmetric), as well as 11 scaliform members plus 54 scaliform fissaries and 2 scaliform compounds.

External links

• Bowers, Jonathan. "Category 18: Ditetrahedrals" (#778).

• Bowers, Jonathan. "How to Make Sidtaxhi".

• Klitzing, Richard. "sidtaxhi".