Pentagrammic retroprism

Pentagrammic retroprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Starp
Coxeter diagram	s2s10/3o ()
Elements
Faces	10 triangles, 2 pentagrams
Edges	10+10
Vertices	10
Vertex figure	Crossed isosceles trapezoid, edge lengths 1, 1, 1, (√5–1)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5-\sqrt5}8} ≈ 0.58779$
Volume	$\frac{5-2\sqrt5}{6} ≈ 0.087977$
Dihedral angles	3–3: $\arccos\left(\frac{\sqrt5}3\right) ≈ 41.81031°$
	5/2–3: $\arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°$
Height	$\sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573$
Central density	3
Number of external pieces	72
Level of complexity	24
Related polytopes
Army	Pap
Regiment	Starp
Dual	Pentagrammic concave antitegum
Conjugate	Pentagonal antiprism
Convex core	Pentagonal antitegum
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	I2(10)×A1/2, order 20
Convex	No
Nature	Tame

The pentagrammic retroprism, or starp, also called the pentagrammic crossed antiprism, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagrams. Each vertex joins one pentagram and three triangles. As the name suggests, it is a crossed antiprism based on a pentagram, seen as a 5/3-gon rather than 5/2. This makes it the simplest uniform crossed antiprism.

Similar to how the pentagonal antiprism can be edge-inscribed into the regular icosahedron, the pentagrammic retroprism can be edge-inscribed into a great icosahedron. It can be constructed by diminishing two opposite vertices of the great icosahedron.

Vertex coordinates

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

• $\left(±\frac{\sqrt5-1}2,\,±\frac12,\,0\right),$
• $\left(0,\,±\frac{\sqrt5-1}2,\,±\frac12\right),$
• $\left(\frac12,\,0,\,\frac{\sqrt5-1}2\right),$
• $\left(-\frac12,\,0,\,-\frac{\sqrt5-1}2\right).$

These coordinates are obtained by removing two opposite vertices from a great icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points along with their central inversions:

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

Related polyhedra

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

• Great inverted snub dodecahedron (6)
• Great inverted disnub dodecahedron (12)

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

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#3 under sissid).

• Klitzing, Richard. "starp".

• Wikipedia Contributors. "Pentagrammic crossed antiprism".
• McCooey, David. "Pentagrammic Crossed Antiprism"