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It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. ... 6: 12: 7: 1: No 4{3}+3{4}
In geometry, a dodecagram (from Greek δώδεκα (dṓdeka) 'twelve' and γραμμῆς (grammēs) 'line' [1]) is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon (with Schläfli symbol {12/5} and a turning number of 5). There are also 4 regular compounds {12/2}, {12/3}, {12/4}, and {12/6}.
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 ...
Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √ φ 6 +2 = √ 8φ+7 for edge length 2. For unit edge length, R must be halved, giving R = √ 8φ+7 / 2 = √ 11+4 √ 5 / 2 ≈ 2.233.
By a theorem of Descartes, this is equal to 4 π divided by the number of vertices (i.e. the total defect at all vertices is 4 π). The three-dimensional analog of a plane angle is a solid angle. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by
It is a polyhedra in a sequence defined by the face configuration V4.6.2n.This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.
All vertices are valence-6 except 12 vertices which are valence 5. Construction of {3,5+} 3,3 Geodesic subdivisions can also be done from an augmented dodecahedron , dividing pentagons into triangles with a center point, and subdividing from that.
Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center. The regular icositetragon has Dih 24 symmetry, order 48. There are 7 subgroup dihedral symmetries: (Dih 12, Dih 6, Dih 3), and (Dih 8, Dih 4, Dih 2 Dih 1), and 8 cyclic group symmetries: (Z 24, Z 12, Z 6, Z 3), and (Z 8 ...