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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 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 ...
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.
Screenshot from Great Stella software, showing the stellation diagram and net for the compound of five tetrahedra Screenshot from Stella4D, looking at the truncated tesseract in perspective and its net, truncated cube cells hidden. Stella is a computer program available in three versions (Great Stella, Small Stella and Stella4D).
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In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5 ⁄ 2,3}. It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex.