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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 ...
5 Uniform nonconvex solids W67 to W119. 6 See also. 7 References. ... Kepler–Poinsot polyhedra (Regular star polyhedra) W20, W21, W22 and W41. Index Name Picture
It is one of five compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It is a stellation of the great icosidodecahedron . It has icosahedral symmetry ( I h ) and it has the same vertex arrangement as a great rhombic triacontahedron .
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 ...
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.
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3, 5 ⁄ 2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
1.4 Kepler-Poinsot solids. 1.5 Achiral nonconvex uniform polyhedra. 2 Chiral Archimedean and Catalan solids. 3 Chiral nonconvex uniform polyhedra. 4 See also.
The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H.S.M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. [91]