Pentagrammic retroprism

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Pentagrammic retroprism
Pentagrammic crossed antiprism.png
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymStarp
Info
Coxeter diagrams2s10/3o
SymmetryI2(10)×A1+, order 20
ArmyPap
RegimentStarp
Elements
Vertex figureCrossed isosceles trapezoid, edge lengths 1, 1, 1, (5–1)/2
Faces10 triangles, 2 pentagrams
Edges10+10
Vertices10
Measures (edge length 1)
Circumradius
Volume
Dihedral angles3–3:
 5/2–3:
Height
Central density3
Euler characteristic2
Related polytopes
DualPentagrammic concave trapezohedron
ConjugatePentagonal antiprism
Convex corePentagonal antitegum
Properties
ConvexNo
OrientableYes
NatureTame

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.

Similar to how the pentagonal antiprism can be vertex-inscribed into the regular icosahedron, the pentagrammic retroprism can be vertex-inscribed into a great icosahedron.

Vertex coordinates[edit | edit source]

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

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:

Related polyhedra[edit | edit source]

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

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

External links[edit | edit source]