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There is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron: it exists as a spherical tiling of digon faces, called a pentagonal hosohedron with Schläfli symbol {2,5}. It has 2 (antipodal point) vertices, 5 edges, and 5 digonal faces.
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
[C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92. [W] Wenninger, 1974, has 119 figures: 1–5 for the Platonic solids, 6–18 for the Archimedean solids, 19–66 for stellated forms including the 4 regular nonconvex polyhedra, and ended ...
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.
Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product {5}×{}. The dual of a pentagonal prism is a pentagonal bipyramid. The symmetry group of a right pentagonal prism is D 5h of order 20. The rotation group is D 5 of order 10.
The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices ...
14.7.5 Tessellations of hyperbolic 3-space. ... and/or vertex figures Abstract Polytopes 1: 1 line ... a face is a facet, an edge is a ridge, and a vertex is a peak ...
It has pentagonal (5 edges) faces, and 3 pentagons around each vertex. See the 5 convex Platonic solids , the 4 nonconvex Kepler-Poinsot polyhedra . Topologically, a regular 2-dimensional tessellation may be regarded as similar to a (3-dimensional) polyhedron, but such that the angular defect is zero.
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