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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 ...
dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex) Tetrahedron {3,3} ... Small stellated dodecahedron ...
3D model of a rhombic dodecahedron. In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces.It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron.
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).
Faces by sides: 20{3}+30{4}+12{5} ... it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, ... preserving angles ...
The pentakis dodecahedron in a model of buckminsterfullerene: each (spherical) surface segment represents a carbon atom, and if all are replaced with planar faces, a pentakis dodecahedron is produced. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.
The critical angle is atan(2) above the dodecahedron face. If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula V − E + F = 2 − 2 g {\displaystyle V-E+F=2-2g} and conclude that the small stellated dodecahedron has genus 4.