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Kepler's final step was to recognize that these polyhedra fit the definition of regularity, even though they were not convex, as the traditional Platonic solids were. In 1809, Louis Poinsot rediscovered Kepler's figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more ...
The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}: As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms.
5 Uniform nonconvex solids W67 to W119. 6 See also. 7 References. ... Kepler–Poinsot polyhedra (Regular star polyhedra) W20, W21, W22 and W41. Index Name Picture
The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polyhedron is dual, or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the ...
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
Kepler–Poinsot solids (regular nonconvex) Small stellated dodecahedron ... Quasiregular dual solids; Rhombic hexahedron (Dual of tetratetrahedron) — V(3.3.3.3)
1.4 Kepler-Poinsot solids. 1.5 Achiral nonconvex uniform polyhedra. 2 Chiral Archimedean and Catalan solids. 3 Chiral nonconvex uniform polyhedra. 4 See also.
Historically, the great dodecahedron is one of two solids discovered by Louis Poinsot in 1810, with some people named it after him, Poinsot solid.As for the background, Poinsot rediscovered two other solids that were already discovered by Johannes Kepler—the small stellated dodecahedron and the great stellated dodecahedron. [3]