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Concept
Snub dodecahedron
Summary
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices.
It has two distinct forms, which are s (or "enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.
Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol and flat Schläfli symbol sr{5,3}.
Let ξ ≈ 0.94315125924 be the real zero of the cubic polynomial x3 + 2x2 − φ2, where φ is the golden ratio. Let the point p be given by
Let the rotation matrices M1 and M2 be given by
and
M1 represents the rotation around the axis (0,1,φ) through an angle of 2pi/5 counterclockwise, while M2 being a cyclic shift of (x,y,z) represents the rotation around the axis (1,1,1) through an angle of 2pi/3. Then the 60 vertices of the snub dodecahedron are the 60 images of point p under repeated multiplication by M1 and/or M2, iterated to convergence. (The matrices M1 and M2 generate the 60 rotation matrices corresponding to the 60 rotational symmetries of a regular icosahedron.) The coordinates of the vertices are integral linear combinations of 1, φ, ξ, φξ, ξ2 and φξ2. The edge length equals
Negating all coordinates gives the mirror image of this snub dodecahedron.
As a volume, the snub dodecahedron consists of 80 triangular and 12 pentagonal pyramids.
The volume V3 of one triangular pyramid is given by:
and the volume V5 of one pentagonal pyramid by:
The total volume is
The circumradius equals
The midradius equals ξ. This gives an interesting geometrical interpretation of the number ξ.
Official source
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