Uniform pentagrammic antiprism | |
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Type | Prismatic uniform polyhedron |
Elements | F = 12, E = 20 V = 10 (χ = 2) |
Faces by sides | 10{3}+2{5/2} |
Schläfli symbol | sr{2,5/2} |
Wythoff symbol | | 2 2 5/2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry | D5h, [5,2], (*552), order 20 |
Rotation group | D5, [5,2]+, (55), order 10 |
Index references | U79(a) |
Dual | Pentagrammic trapezohedron |
Properties | nonconvex |
![]() Vertex figure 3.3.3.5/2 |
In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Pentagrammic_antiprism.stl/220px-Pentagrammic_antiprism.stl.png)
It has 12 faces, 20 edges and 10 vertices. This polyhedron is identified with the indexed name U79 as a uniform polyhedron.[1]
Note that the pentagram face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior depending on how interior is defined. One definition of interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the perimeter.
In either case, it is best to show the pentagram boundary line to distinguish it from a concave decagon.
Gallery
editAn alternative representation with hollow centers to the pentagrams. | The pentagrammic trapezohedron is the dual to the pentagrammic antiprism. |
Net
editNet (fold the dotted line in the centre in the opposite direction to all the other lines):
See also
editReferences
edit- ^ Maeder, Roman. "79: pentagrammic antiprism".
External links
edit- Weisstein, Eric W. "Pentagrammic antiprism". MathWorld.
- http://www.mathconsult.ch/showroom/unipoly/04.html
- https://web.archive.org/web/20050313233653/http://www.math.technion.ac.il/~rl/kaleido/data/04.html