Snub dodecahedron

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The snub dodecahedron (also called the snub icosidodecahedron) is an Archimedean solid—one of the 13 convex isogonal solids made from two or more types of regular polygon faces. It is chiral, meaning it has two mirror-image forms (left-handed and right-handed). These two enantiomorphs together form a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.

Its faces total 92: 12 regular pentagons and 80 equilateral triangles. It has 150 edges and 60 vertices. The snub dodecahedron is vertex-transitive but not uniform, and its rotational symmetry group has 60 elements, the same order as the rotational symmetries of an icosahedron.

There are two common ways to construct it. One starts with a regular dodecahedron: the pentagonal faces are pulled outward and gaps are filled with triangles in a specific way. For the snub form, the pentagons are rotated a little and triangles are added, while leaving some gaps empty so that only triangles fill the gaps, producing the snub pattern. The other construction uses alternation from a truncated icosidodecahedron: selecting alternating vertices yields a snub dodecahedron, while the remaining alternating set gives its mirror image.

Geometrically, the snub dodecahedron has interesting measurements. If its edge length is set to 1, its surface area is about 55.287 and its volume about 37.617. It also contains two inscribed spheres: one tangent to all triangular faces with radius around 2.077, and a slightly smaller one tangent to all pentagonal faces with radius around 1.981. Its two inscribed spheres and edge/face structure arise from the same underlying golden-ratio geometry that governs many related polyhedra.

The snub dodecahedral graph, formed by its vertices and edges, has 60 vertices and 150 edges. It also has the distinction of having one of the highest sphericities among Archimedean solids, indicating its shape is very close to that of a perfect sphere.