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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 dihedral angle of a truncated icosahedron between adjacent hexagonal faces is approximately 138.18°, and that between pentagon-to-hexagon is approximately 142.6°. [ 4 ] The truncated icosahedron is an Archimedean solid , meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in ...
Therefore, the dihedral angle of a cuboctahedron between square-to-triangle is approximately 125°. [7] The process of jitterbug transformation. Buckminster Fuller found that the cuboctahedron is the only polyhedron in which the distance between its center to the vertex is the same as the distance between its edges. In other words, it has the ...
Its dihedral angles can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following: [4] its dihedral angle between triangle and hexagon is as in the truncated tetrahedron: 109.47°; its dihedral angle between adjacent hexagons is as in the truncated tetrahedron: 70.53°;
In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size. The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic ...