Quasiprismatorhombated faceted hexacosichoron

Quasiprismatorhombated faceted hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Quippirfix
Coxeter diagram	x5/3x5o3x ()
Elements
Cells	1200 triangular prisms, 720 decagrammic prisms, 120 quasitruncated small stellated dodecahedra, 120 small rhombicosidodecahedra
Faces	2400 triangles, 3600+3600 squares, 1440 pentagons, 1440 decagrams
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths 1, √2, (1+√5)/2, √2; lateral edge lengths √2, √2, √(5–√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {8+2{\sqrt {5}}}}\approx 3.53159$
Hypervolume	$25{\frac {511+163{\sqrt {5}}}{4}}\approx 5471.74425$
Dichoral angles	Stiddip–4–trip: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	Quit sissid–10/3–stiddip: 162°
	Srid–5–quit sissid: 36°
	Srid–4–stiddip: $\arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }$
	Srid–3–trip: 30°
Related polytopes
Army	Semi-uniform Prahi, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (small rhombicosidodecahedra), ${\frac {{\sqrt {5}}-1}{2}}$ (sides of pentagonal prisms)
Regiment	Quippirfix
Conjugate	Prismatoquasirhombated great faceted hexacosichoron
Abstract & topological properties
Flag count	230400
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The quasiprismatorhombated faceted hexacosichoron, or quippirfix, is a nonconvex uniform polychoron that consists of 1200 triangular prisms, 720 decagrammic prisms, 120 quasitruncated small stellated dodecahedra, and 120 small rhombicosidodecahedra. 1 triangular prism, 2 decagrammic prisms, 1 small rhombicosidodecahedron, and 1 quasitruncated small stellated dodecahedron join at each vertex. It can be obtained by quasiruncitruncating the small stellated hecatonicosachoron.

Cross-sections

Vertex coordinates

The vertices of a quasiprismatorhombated faceted hexacosichoron of edge length 1 are given by all permutations of:

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

Plus all even permutations of:

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

Related polychora

The quasiprismatorhombated faceted hexacosichoron is the colonel of a 3-member regiment that also includes the great prismatotrishecatonicosachoron and the rhombiprismic dishecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#382).

• Klitzing, Richard. "quippirfix".