Pentagonal-prismatic heptacosiicosachoron

Pentagonal-prismatic heptacosiicosachoron
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Paphacki
Elements
Cells	720 pentagonal prisms
Faces	1800 squares, 720 pentagons
Edges	1200
Vertices	120
Vertex figure	Great noble triangular hexecontahedron
Edge figure	(pip 4 pip 4 pip 5)×3
Measures (edge length 1)
Circumradius	$1$
Inradius	$\frac{\sqrt{25-10\sqrt5}}{10} \approx 0.16246$
Volume	$\frac{45\sqrt5}{2} \approx 50.31153$
Number of external pieces	1260000
Level of complexity	3490
Related polytopes
Army	Ex
Regiment	Sishi
Conjugate	Pentagrammic-prismatic heptacosiicosachoron
Convex core	Joined hecatonicosachoron
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Feral

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

Vertex coordinates

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

• $\left(±1,\,0,\,0,\,0\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right),$

and all even permutations of:

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

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" (#769).
• Klitzing, Richard. "Paphacki".