Snub dodecahedron
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 : 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 mirror images 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.
Construction
The snub dodecahedron can be constructed from a regular dodecahedron, separating all of its pentagonal faces and filling the gaps with equilateral triangles. The name originates from Kepler's Harmonices Mundi in 1619, written in the language of Latin as dodecahedron simum. H. S. M. Coxeter noted it could be derived equally from either the regular octahedron or the icosahedron.With the regular dodecahedron, the snub dodecahedron can be generated from taking the twelve pentagonal faces and pulling them outward so the faces can no longer touch. At a proper distance, this can create the rhombicosidodecahedron by filling in square faces between the divided edges and triangle faces between the divided vertices. But for the snub form, pull the pentagonal faces out slightly less, only add the triangle faces, and leave the other gaps empty. Then apply an equal rotation to the centers of the pentagons and triangles, continuing the rotation until the gaps can be filled by two equilateral triangles.
The snub dodecahedron can also be derived from the truncated icosidodecahedron by the process of alternation. Sixty of the vertices of the truncated icosidodecahedron form a polyhedron topologically equivalent to one snub dodecahedron; the remaining sixty form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform.
Cartesian coordinates
Let be the real zero of the cubic polynomial, where is the golden ratio. Let the point be given byLet the rotation matrices and be given by
represents the rotation around the axis through an angle of counterclockwise, while being a cyclic shift of represents the rotation around the axis through an angle of. Then the 60 vertices of the snub dodecahedron are the 60 images of point under repeated multiplication by and/or, iterated to convergence. The coordinates of the vertices are integral linear combinations of and. The edge length equals
Negating all coordinates gives the mirror image of this snub dodecahedron.
Properties
Measurement
For a snub dodecahedron whose edge length is 1, the surface area isIts volume is
There are two inscribed spheres, one touching the triangular faces and one, slightly smaller, touching the pentagonal faces. Their radii are, respectively:
Alternatively, r3 and r5 can be expressed as roots of the following respective polynomials:
The four positive real roots of the sextic equation in R2are the circumradii of the snub dodecahedron, great snub icosidodecahedron, great inverted snub icosidodecahedron, and great retrosnub icosidodecahedron.
Sphericity
The snub dodecahedron has the highest sphericity of all Archimedean solids. If sphericity is defined as the ratio of volume squared over surface area cubed, multiplied by a constant of 36, the sphericity of the snub dodecahedron is about 0.947.Geometric relations
Alternatively, combining the vertices of the snub dodecahedron given by the Cartesian coordinates and its mirror will form a semiregular truncated icosidodecahedron. The comparisons between these regular and semiregular polyhedrons is shown in the figure to the right.Cartesian coordinates for the vertices of this alternative snub dodecahedron are obtained by selecting sets of 12. The alternations are those with an odd number of minus signs in these three sets:
and an even number of minus signs in these two sets:
where is the golden ratio. The mirrors of both the regular truncated icosidodecahedron and this alternative snub dodecahedron are obtained by switching the even and odd references to both sign and position permutations.