Pentagrammic prism

Summary

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Uniform pentagrammic prism
Type Prismatic uniform polyhedron
Elements F = 7, E = 15
V = 10 (χ = 2)
Faces by sides 5{4}+2{5/2}
Schläfli symbol t{2,5/2} or {5/2}×{}
Wythoff symbol 2 5/2 | 2
Coxeter diagram
Symmetry D5h, [5,2], (*522), order 20
Rotation group D5, [5,2]+, (522), order 10
Index references U78(a)
Dual Pentagrammic dipyramid
Properties nonconvex

Vertex figure
4.4.5/2

In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.

It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex pentagonal prism.

It is the 78th model in the list of uniform polyhedra, as the first representative of uniform star prisms, along with the pentagrammic antiprism, which is the 79th model.

Geometry

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It has 7 faces, 15 edges and 10 vertices. This polyhedron is identified with the indexed name U78 as a uniform polyhedron.[1]

The triangle face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered an angle or exterior depending on how the interior is defined. One definition of the interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the diameter

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An alternative representation with hollow centers to the pentagrams.
 
3D model of a (uniform) pentagrammic prism

Pentagrammic dipyramid

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Pentagrammic dipyramid
 
TypeStar bipyramid
Faces10 triangles
Edges15
Vertices7
Schläfli symbol{} + {5/2}
Coxeter diagram       
Symmetry groupD5h, [5,2], (*225), order 20
Rotation groupD5, [5,2]+, (225), order 10
Dual polyhedronpentagrammic prism
Face configurationV4.4.5
Propertiesface-transitive, (deltahedron)

In geometry, the pentagrammic dipyramid (or bipyramid) is first of the infinite set of face-transitive star dipyramids containing star polygon arrangement of edges. It has 10 intersecting isosceles triangle faces. It is topologically identical to the pentagonal dipyramid.

Each star dipyramid is the dual of a star polygon based uniform prism.

 
3D model of a dual uniform pentagrammic dipyramid
 
3D model of a pentagrammic dipyramid with regular faces
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There are two pentagrammic trapezohedra (or deltohedra), being dual to the pentagrammic antiprism and pentagrammic crossed antiprism respectively, each having intersecting kite-shaped faces (convex or concave), and a total of 12 vertices:

{52} trapezohedron {53} trapezohedron
   

References

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  1. ^ Maeder, Roman. "78: pentagrammic prism". MathConsult.
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  • Weisstein, Eric W. "Pentagrammic prism". MathWorld.
  • Weisstein, Eric W. "Pentagrammic dipyramid". MathWorld.
  • Weisstein, Eric W. "Pentagrammic deltohedron". MathWorld.
  • Weisstein, Eric W. "Pentagrammic concave deltohedron". MathWorld.
  • http://www.mathconsult.ch/showroom/unipoly/78.html
  • http://bulatov.org/polyhedra/uniform/u03.html
  • Paper model of pentagrammic prism
  • https://web.archive.org/web/20050313234702/http://www.math.technion.ac.il/~rl/kaleido/data/03.html
  • https://web.archive.org/web/20060211140715/http://www.ac-noumea.nc/maths/amc/polyhedr/no_conv5_.htm
  • Paper Model (net) Pentagrammic Prism