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The 5 Platonic solids are called a ... 7: 5{4} +2{5} Hexagonal prism: 4.4.6: ... The white polygon lines represent the "vertex figure" polygon. The colored faces are ...
The quantity h (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively. The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2 π .
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.
Vertex the (n−5)-face of the 5-polytope; Edge the (n−4)-face of the 5-polytope; Face the peak or (n−3)-face of the 5-polytope; Cell the ridge or (n−2)-face of the 5-polytope; Hypercell or Teron the facet or (n−1)-face of the 5-polytope
Apollonian gasket; Apollonian sphere packing; Blancmange curve; Cantor dust; Cantor set; Cantor tesseract [citation needed]; Circle inversion fractal; De Rham curve; Douady rabbit; Dragon curve
where φ = 1 + √ 5 / 2 is the golden ratio. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √ φ 6 +2 = √ 8φ+7 for edge length 2. For unit edge length, R must be halved, giving R = √ 8φ+7 / 2 = √ 11+4 √ 5 / 2 ≈ 2.233.
[3] [5] [6] A face of dimension k is called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. For any n-polytope (n-dimensional polytope), −1 ≤ k ≤ n.
These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners. The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον ( polygōnon/polugōnon ), noun use of neuter of πολύγωνος ( polygōnos/polugōnos , the masculine ...