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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 ...
It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes. It contains the 75 nonprismatic uniform polyhedra , as well as 44 stellated forms of the convex regular and quasiregular polyhedra.
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 ...
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]
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.
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
That is, it is correct and appropriate to say that a Platonic solid or regular polyhedron or Kepler–Poinsot polyhedron is defined by its property of being flag-transitive (with the only difference between these three classes being what you think of as a polyhedron, and with other variants on abstract manifolds that are also called regular ...