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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 .
A dihedral angle is the angle between two intersecting planes or half-planes. It is a plane angle formed on a third plane, perpendicular to the line of intersection between the two planes or the common edge between the two half-planes. In higher dimensions, a dihedral angle represents the angle between two hyperplanes.
For example, with 4 square faces, and 60-degree rhombic faces, and D 4h dihedral symmetry, order 16. It can be seen as a cuboctahedron with square pyramids attached on the top and bottom. In 1960, Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. It has the same topology but ...
For instance, for the ideal cube, the dihedral angles are / and their supplements are /. The three supplementary angles at a single vertex sum to 2 π {\displaystyle 2\pi } but the four angles crossed by a curve midway between two opposite faces sum to 8 π / 3 > 2 π {\displaystyle 8\pi /3>2\pi } , and other curves cross even more of these ...
The 12 face angles - there are three of them for each of the four faces of the tetrahedron. The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge. The 4 solid angles - associated to each point of the tetrahedron.
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 ...
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 ...
The dihedral angle equals (+). The ratio between the lengths of the long edges and the short ones equals +, which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.