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In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. [ 1 ] They may be obtained by stellating the regular convex dodecahedron and icosahedron , and differ from these in having regular pentagrammic faces or vertex figures .
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).
Historically, the great dodecahedron is one of two solids discovered by Louis Poinsot in 1810, with some people named it after him, Poinsot solid.As for the background, Poinsot rediscovered two other solids that were already discovered by Johannes Kepler—the small stellated dodecahedron and the great stellated dodecahedron. [3]
The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, ... Edge framework of cubic honeycomb, {4,3,4} ... 0 {4,4,3} Square ...
Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of ...
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.
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:
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.