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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)
In pyritohedral pyrite, the faces have a Miller index of (210), which means that the dihedral angle is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the ...
A regular dodecahedron or pentagonal dodecahedron [notes 1] is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron ...
This dihedral angle, also called the face angle, is measured as the internal angle with respect to the polyhedron. 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 ...
The stellation diagram for the regular dodecahedron with the central pentagon highlighted. This diagram represents the dodecahedron face itself. In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one.
The regular dodecahedron can be faceted into one regular Kepler–Poinsot polyhedron, three uniform star polyhedra, and three regular polyhedral compound. The uniform stars and compound of five cubes are constructed by face diagonals. The excavated dodecahedron is a facetting with star hexagon faces.
The rhombic dodecahedron can be decomposed into six congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of six pairs of the 24-cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron.
A snub cube can be constructed from a rhombicuboctahedron by rotating the 6 blue square faces until the 12 white square faces become pairs of equilateral triangle faces. In geometry , a snub is an operation applied to a polyhedron .