Medial prismatotrishecatonicosachoron

Medial prismatotrishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Mipthi
Coxeter diagram	x5x3o5/2x5/3*b ()
Elements
Cells	720 pentagrammic prisms, 120 truncated dodecahedra, 120 great dodecicosidodecahedra, 120 quasitruncated dodecadodecahedra
Faces	2400 triangles, 3600 squares, 1440 pentagrams, 1440 decagons, 1440 decagrams
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths 1, √(5–√5)/2, (√5–1)/2, √(5–√5)/2; lateral edge lengths √2, √2, √(5+√5)/2, √(5+√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {8+{\sqrt {5}}}}\approx 3.19939$
Hypervolume	$5{\frac {1905+913{\sqrt {5}}}{4}}\approx 4933.16258$
Dichoral angles	Gaddid–5/2–stip: 162°
	Quitdid–4–stip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	Gaddid–10/3–quitdid: 144°
	Tid–10–quitdid: 72°
	Gaddid–3–tid: 60°
Related polytopes
Army	Semi-uniform Grix, edge lengths ${\sqrt {5}}-2$ (pentagons), ${\frac {3-{\sqrt {5}}}{2}}$ (remaining edges of truncated icosahedra), ${\frac {{\sqrt {5}}-1}{2}}$ (sides of pentagonal prisms)
Regiment	Mipthi
Conjugate	Grand prismatotrishecatonicosachoron
Abstract & topological properties
Flag count	230400
Euler characteristic	–1560
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The medial prismatotrishecatonicosachoron, or mipthi, is a nonconvex uniform polychoron that consists of 720 pentagrammic prisms, 120 truncated dodecahedra, 120 great dodecicosidodecahedra, and 120 quasitruncated dodecadodecahedra. 1 pentagrammic prism, 1 truncated dodecahedron, 1 great dodecicosidodecahedron, and 2 quasitruncated dodecadodecahedra join at each vertex.

Vertex coordinates

The vertices of a medial prismatotrishecatonicosachoron of edge length 1 are given by all permutations of:

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

Plus all even permutations of:

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

Related polychora

The medial prismatotrishecatonicosachoron is the colonel of a three-member ergiment that also includes the prismatoquasirhombated great stellated hecatonicosachoron and the medial rhombiprismic hecatonicosihecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#400).

• Klitzing, Richard. "mipthi".