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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.
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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).
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.
The net of Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler–Poinsot polyhedra. [15]
Políedre de Kepler-Poinsot; Usage on cs.wikipedia.org Wikipedista diskuse:Glivi/Archiv do 5.3. 2007; Usage on fi.wikipedia.org Keplerin–Poinsot’n kappale; Usage on fr.wikipedia.org Polyèdre; Usage on gl.wikipedia.org Poliedro regular; Usage on ko.wikipedia.org 케플러-푸앵소 다면체; Usage on oc.wikipedia.org Solids de Kepler-Poinsot
And I don't see why Poinsot's solids should receive a different treatment from Kepler's, to be honest; the great dodecahedron was also anticipated before Kepler and Poinsot. (The great icosahedron alone seems not to have been anticipated, as it is the only one that cannot be obtained directly from augmenting or excavating {5, 3} and {3, 5}.)