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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.
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 ...
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.
Kepler (1619) discovered two of the regular Kepler–Poinsot polyhedra, the small stellated dodecahedron and great stellated dodecahedron. Louis Poinsot (1809) discovered the other two, the great dodecahedron and great icosahedron. The set of four was proven complete by Augustin-Louis Cauchy in 1813 and named by Arthur Cayley in 1859.
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]
3 Kepler–Poinsot polyhedra (Regular star polyhedra) W20, W21, W22 and W41. 4 Stellations: models W19 to W66. ... 5 Uniform nonconvex solids W67 to W119. 6 See also ...
5 Platonic solids: 4 Kepler–Poinsot solids: 3 tilings: ... Kepler–Poinsot polyhedron (regular star polyhedra) Great icosahedron; Small stellated dodecahedron;