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Picture Name Schläfli symbol Vertex/Face configuration exact dihedral angle (radians) dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex)
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 ...
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 property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of a convex regular polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal; All the vertex figures of the polyhedron are regular polygons.
The dihedral angle of an icosidodecahedron between pentagon-to-triangle is (+), determined by calculating the angle of a pentagonal rotunda. [ 4 ] An icosidodecahedron has icosahedral symmetry , and its first stellation is the compound of a dodecahedron and its dual icosahedron , with the vertices of the icosidodecahedron located at the ...
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 ...
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
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 ...