Small disprismatohexacosihecatonicosachoron

Small disprismatohexacosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Sidpixhi
Coxeter diagram	x5o3o3x ()
Elements
Cells	600 tetrahedra, 1200 triangular prisms, 720 pentagonal prisms, 120 dodecahedra
Faces	2400 triangles, 3600 squares, 1440 pentagons
Edges	3600+3600
Vertices	2400
Vertex figure	Triangular antipodium, edge lengths 1 (small base), (1+√5)/2 (large base), and √2 (sides)
Measures (edge length 1)
Circumradius	$3+{\sqrt {5}}\approx 5.23607$
Hypervolume	$25{\frac {281+126{\sqrt {5}}}{4}}\approx 3517.15353$
Dichoral angles	Tet–3–trip: $\arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }$
	Pip–4–trip: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	Doe–5–pip: 162°
Central density	1
Number of external pieces	2640
Level of complexity	8
Related polytopes
Army	Sidpixhi
Regiment	Sidpixhi
Dual	Triangular-antitegmatic dischiliatetracosichoron
Conjugate	Quasidisprismatohexacosihecatonicosachoron
Abstract & topological properties
Flag count	115200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

The small disprismatohexacosihecatonicosachoron, or sidpixhi, also commonly called the runcinated 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra, 120 regular dodecahedra, 1200 triangular prisms, and 720 pentagonal prisms. 1 tetrahedron, 1 dodecahedron, 3 triangular prisms, and 3 pentagonal prisms join at each vertex. It is the result of expanding the cells of either a hecatonicosachoron or a hexacosichoron outwards, and thus could also be called the runcinated 600-cell.

Cross-sections

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)$,

along with the even permutations of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)$,
• $\left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)$.

Semi-uniform variant

The small disprismatohexacosihecatonicosachoron has a semi-uniform variant of the form x5o3o3y that maintains its full symmetry. This variant uses 120 dodecahedra of size x, 600 tetrahedra of size y, 1200 semi-uniform triangular prisms of form x y3o, and 720 semi-uniform pentagonal prisms of form y x5o as cells, with 2 edge lengths.

With edges of length a (of dodecahedra) and b (of tetrahedra), its circumradius is given by ${\sqrt {\frac {14a^{2}+3b^{2}+11ab+(6a^{2}+b^{2}+5ab){\sqrt {5}}}{2}}}$.

Related polychora

The small disprismatohexacosihecatonicosachoron is the colonel of a 7-member regiment.

Sidpixhi regiment

Index	Name	OBSA	Company	Nature
454	Small disprismatohexacosihecatonicosachoron	Sidpixhi	Sidpixhi	Tame
455	Small hecatonicosafaceted prismatohecatonicosihexacosichoron	Shif Phix	Sidpixhi	Tame
456	Small hexacosifaceted prismatodishecatonicosachoron	Six Fipady	Sidpixhi	Tame
457	Small hexacosihecatonicosihecatonicosachoron	Sixhihy	Sixhihy	Tame
458	Prismatoprismatohecatonicosachoron	Paphi		Wild
459	Small spinoprismatohexacosihecatonicosachoron	Snapaxhi		Wild
460	Small spinoprismatodishecatonicosachoron	Snappady		Wild

The segmentochoron dodecahedron atop small rhombicosidodecahedron can be obtained as a cap of the small disprismatohexacosihecatonicosachoron. Another segmentochoral cap is the pentagonal cupofastegium, when seen pentagonal prism-first.

External links

• Bowers, Jonathan. "Category 11: Antipodiumverts" (#454).

• Klitzing, Richard. "sidpixhi".
• Quickfur. "The Runcinated 120-cell".

• Wikipedia contributors. "Runcinated 120-cell".