Pentagrammic-prismatic heptacosiicosachoron

Pentagrammic-prismatic heptacosiicosachoron
Rank	4
Type	Uniform
Notation
Bowers style acronym	Paphicki
Elements
Cells	720 pentagrammic prisms
Faces	1800 squares, 720 pentagrams
Edges	1200
Vertices	120
Vertex figure	Small noble triangular hexecontahedron
Edge figure	(stip 4 stip 4 stip 5/2)×3
Measures (edge length 1)
Circumradius	$1$
Inradius	${\frac {\sqrt {25+10{\sqrt {5}}}}{10}}\approx 0.68819$
Volume	${\frac {45{\sqrt {5}}}{2}}\approx 50.31153$
Central density	29
Number of external pieces	8640
Level of complexity	16
Related polytopes
Army	Ex
Regiment	Sishi
Conjugate	Pentagonal-prismatic heptacosiicosachoron
Convex core	Joined hecatonicosachoron
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Feral

The pentagrammic-prismatic heptacosiicosachoron, or paphicki, is a noble uniform polychoron in the small stellated hecatonicosachoron's regiment. It consists of 720 pentagrammic prisms, 60 of which meet each of its 120 vertices.

Vertex coordinates

The vertices of a pentagrammic-prismatic heptacosiicosachoron of unit edge length are given by:

• $\left(\pm 1,\,0,\,0,\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$

and all even permutations of:

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

The first two sets of vertices form a unit icositetrachoron that can be inscribed into the small stellated hecatonicosachoron. This corresponds to the fact there is a compound of 25 icositetrachora with the same vertices and edges as the small stellated hecatonicosachoron.

External links

• Bowers, Jonathan. "Category 17: Sishi Regiment" (#768).
• Klitzing, Richard. "Paphicki".