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.
It has 12 faces, 20 edges and 10 vertices. This polyhedron is identified with the indexed name U79 as a uniform polyhedron.
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.
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In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids. Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group.
In geometry, the pentagrammic crossed-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. It differs from the pentagrammic antiprism by having opposite orientations on the two pentagrams. This polyhedron is identified with the indexed name U80 as a uniform polyhedron. The pentagrammic crossed-antiprism may be inscribed within an icosahedron, and has ten triangular faces in common with the great icosahedron.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.