Small ditetrahedronary hexacosihecatonicosachoron

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.

Small ditetrahedronary hexacosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sidtaxhi
Coxeter diagram	o3o3x5/2o3*b ()
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 external pieces	2520
Level of complexity	9
Related polytopes
Army	Hi, edge length $\frac{\sqrt5-1}{2}$
Regiment	Sidtaxhi
Conjugate	Grand ditetrahedronary hexacosihecatonicosachoron
Abstract & topological properties
Flag count	43200
Euler characteristic	—600
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

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".