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Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Table I: Regular Polytopes, (i) The nine regular polyhedra {p,q} in ordinary space) Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design .
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 ...
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 solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by Ω = q θ − ( q − 2 ) π . {\displaystyle \Omega =q\theta -(q-2)\pi .\,} This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron { p , q } is a regular q -gon.
the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a regular octagon 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is ...
The exsphere touches the face of the regular polyedron at the center of the incircle of that face. If the exsphere radius is denoted r ex, the radius of this incircle r in and the dihedral angle between the face and the extension of the adjacent face δ, the center of the exsphere is located from the viewpoint at the middle of one edge of the face by bisecting the dihedral angle.
The dihedral angle of a triangular cupola between square-to-triangle is approximately 125°, that between square-to-hexagon is 54.7°, and that between triangle-to-hexagon is 70.5°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle, on the edge where the base of two triangular cupolas are attached is 54.7° + 70.5 ...