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In geometry, the complete or final stellation of the icosahedron [1] is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or ...
Before Coxeter, only Brückner and Wheeler had recorded any significant sets of stellations, although a few such as the great icosahedron had been known for longer. Since publication of The 59 , Wenninger published instructions on making models of some; the numbering scheme used in his book has become widely referenced, although he only ...
There are 58 stellations of the icosahedron, including the great icosahedron (one of the Kepler–Poinsot polyhedra), and the second and final stellations of the icosahedron. The 59th model in The fifty nine icosahedra is the original icosahedron itself. Many "Miller stellations" cannot be obtained directly by using Kepler's method.
Icosahedron: Small triambic icosahedron: Icosahedron: Great triambic icosahedron: Icosahedron: Compound of five cubes: Rhombic triacontahedron: Compound of great icosahedron and great stellated dodecahedron: Icosidodecahedron: Compound of great icosahedron and great stellated dodecahedron: Great icosidodecahedron: Compound of dodecahedron and ...
In this way he constructed the two stellated dodecahedra. Each has the central convex region of each face "hidden" within the interior, with only the triangular arms visible. 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.
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes.
The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of six or twelve pentagrammic prisms. The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced ...
The regular octahedron is dual to the cube, and the regular icosahedron is dual to the regular dodecahedron. The regular tetrahedron is self-dual, meaning its dual is the regular tetrahedron itself. [1] 4 regular star Kepler–Poinsot solids: great dodecahedron, small stellated dodecahedron, great icosahedron, and great stellated dodecahedron ...