Small tritrigonary prismatohecatonicosidishexacosichoron

Small tritrigonary prismatohecatonicosidishexacosichoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Stut phiddix
Coxeter diagram	x3o3o5/2x3*b ()
Elements
Cells	600 tetrahedra, 720 pentagrammic prisms, 600 cuboctahedra, 120 small ditrigonary icosidodecahedra
Faces	2400+2400 triangles, 3600 squares, 1440 pentagrams
Edges	3600+7200
Vertices	2400
Vertex figure	Triangular cupola, edge lengths 1 (top triangle and 3 edges of base ditrigon), (√5–1)/2 (remaining base edges), and √2 (side edges)
Measures (edge length 1)
Circumradius	$\sqrt{8+3\sqrt5} ≈ 3.83513$
Hypervolume	$5\frac{960+281\sqrt5}{4} ≈ 1985.41888$
Dichoral angles	Co–4–stip: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) ≈ 166.71747°$
	Tet–3–co: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751°$
	Sidtid–5/2–stip: 162°
	Sidtid–3–co: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124°$
Number of external pieces	3840
Related polytopes
Army	Semi-uniform sidpixhi
Regiment	Stut phiddix
Conjugate	Great tritrigonary prismatohecatonicosidishexacosichoron
Convex core	Semi-uniform grix
Abstract & topological properties
Euler characteristic	–600
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small tritrigonary prismatohecatonicosidishexacosichoron, or stut phiddix, is a nonconvex uniform polychoron that consists of 600 regular tetrahedra, 720 pentagrammic prisms, 600 cuboctahedra, and 120 small ditrigonary icosidodecahedra. 1 tetrahedron, 3 pentagrammic prisms, 3 cuboctahedra, and 1 small ditrigonary icosidodecahedron join at each vertex.

It can be constructed from the small ditetrahedronary hexacosihecatonicosachoron by expanding its small ditrigonary icosidodecahedral cells outward.

The small tritrigonary prismatohecatonicosidishexacosichoron contains the vertices of an inscribed great rhombicosidodecahedral prism.

Vertex coordinates

The vertices of a small tritrigonary prismatohecatonicosidishexacosichoron of edge length 1 are all permutations of:

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

along with the even permutations of:

• $\left(0,\,±\frac{\sqrt5-1}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{3+2\sqrt5}{2}\right),$
• $\left(0,\,±\frac12,\,±\frac{5+3\sqrt5}{4},\,±3\frac{1+\sqrt5}{4}\right),$
• $\left(0,\,±\frac{\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$
• $\left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{4+\sqrt5}{2}\right),$
• $\left(±\frac{\sqrt5-1}{4},\,±\frac12,\,±\frac{7+3\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac{\sqrt5-1}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac12,\,±1,\,±\frac{7+3\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4}\right).$

Related polychora

The small tritrigonary prismatohecatonicosidishexacosichoron is the colonel of a regiment that contains 79 uniform members as well as 30 fissary uniforms.

External links

• Bowers, Jonathan. "Category 24: Stut Phiddix Regiment" (#1304).

• Bowers, Jonathan. "How to Make Stut Phiddix".

• Klitzing, Richard. "stut phiddix".