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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 ...
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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]
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).
Analogous to the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular Platonic solids or one of the four regular star Kepler–Poinsot polyhedra. For example, the great grand stellated 120-cell, projected orthogonally into 3-space, looks like this:
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}.)