Pentagrammic prism

The pentagrammic prism, or stip, is a prismatic uniform polyhedron. It consists of 2 pentagrams and 5 squares. Each vertex joins one pentagram and two squares. As the name suggests, it is a prism based on a pentagram.

Pentagrammic prism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Stip
Coxeter diagram	x x5/2o ()
Elements
Faces	5 squares, 2 pentagrams
Edges	5+10
Vertices	10
Vertex figure	Isosceles triangle, edge lengths (√5–1)/2, √2, √2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{15-2\sqrt5}{20}} ≈ 0.72553$
Volume	$\frac{\sqrt{25-10\sqrt5}}{4}≈ 0.40615$
Dihedral angles	4–5/2: 90°
	4–4: 36°
Height	1
Central density	2
Number of external pieces	12
Level of complexity	6
Related polytopes
Army	Semi-uniform Pip
Regiment	Stip
Dual	Pentagrammic tegum
Conjugate	Pentagonal prism
Convex core	Pentagonal prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H2×A1, order 20
Convex	No
Nature	Tame

Vertex coordinates

A pentagrammic prism of edge length 1 has vertex coordinates given by:

• $\left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,±\frac12\right),$
• $\left(±\frac{\sqrt5-1}{4},\,\sqrt{\frac{5+\sqrt5}{40}},\,±\frac12\right),$
• $\left(0,\,-\sqrt{\frac{5-\sqrt5}{10}},\,±\frac12\right).$

Related polyhedra

Two non-prismatic uniform polyhedron compounds are composed of pentagrammic prisms:

• Great chirorhombidodecahedron (6)
• Great disrhombidodecahedron (12)

There are also an infinite amount of prismatic uniform compounds that are the prisms of compounds of pentagrams.

External links

• Klitzing, Richard. "stip".

• Wikipedia Contributors. "Pentagrammic prism".
• McCooey, David. "Pentagrammic Prism"