In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.
It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron.
It is considered the first of three stellations of the dodecahedron.
If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar.
Contents
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Images 1
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In art 2
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Related polyhedra 3
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References 4
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See also 5
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External links 6
Images
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Transparent model
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Handmade models
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(See also: )
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Spherical tiling
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Stellation
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Net
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This polyhedron also represents a spherical tiling with a density of 3. (One spherical pentagram face, outlined in blue, filled in yellow)
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It can also be constructed as the first of three stellations of the dodecahedron, and referenced as Wenninger model [W20].
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Small stellated dodecahedra can be constructed out of paper or cardstock by connecting together 12 five-sided isosceles pyramids in the same manner as the pentagons in a regular dodecahedron. With an opaque material, this visually represents the exterior portion of each pentagrammic face.
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In art
Floor mosaic by Paolo Uccello, 1430
Related polyhedra
Its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron.
This polyhedron is the truncation of the great dodecahedron:
The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping (as truncated pentagrams).
References
See also
External links
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Uniform polyhedra and duals
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Bronze sculpture of small stellated dodecahedron
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