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The dihedral angles for the edge-transitive polyhedra are: Picture Name Schläfli symbol Vertex/Face configuration exact dihedral angle (radians) dihedral angle ...
An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling. An angle greater than 180° exists on concave portions of a polyhedron. Every dihedral angle in an edge-transitive polyhedron ...
A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In ...
This fact can be used to calculate the dihedral angles themselves for a regular or edge-symmetric ideal polyhedron (in which all these angles are equal), by counting how many edges meet at each vertex: an ideal regular tetrahedron, cube or dodecahedron, with three edges per vertex, has dihedral angles = / = (), an ideal regular octahedron or ...
Its dihedral angle between two rhombi is 120°. [2] The rhombic dodecahedron is a Catalan solid, meaning the dual polyhedron of an Archimedean solid, the cuboctahedron; they share the same symmetry, the octahedral symmetry. [2] It is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces.
The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges. [44] Another of Hilbert's problems, Hilbert's 18th problem, concerns (among other things) polyhedra that tile space. Every such polyhedron must have Dehn invariant zero. [45]
Determine the edge lengths and dihedral angles (the angle between two faces meeting along an edge) of all of the polyhedra. Find a subset of the angles that forms a rational basis. This means that each dihedral angle can be represented as a linear combination of basis elements, with rational number coefficients.
The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. [1] It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. [2] This set of polyhedrons is named after Plato.