Symmetry Groups of Platonic Solids
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منابع مشابه
The Platonic Solids
The tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. From a first glance, one immediately notices that the Platonic Solids exhibit remarkable symmetry. They are the only convex polyhedra for which the same same regular polygon is used for each face, and the same number of faces meet at each vertex. Their symmetries are aesthetically pleasing, like those of stones cu...
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In this note we will determine the symmetry groups of the Platonic solids. We will use Maple to help us do this. The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. We will view the symmetry groups of these solids in two ways. By definition, the symmetry group of a solid is the set of isometries of R that stabilize the set. In each of t...
متن کاملR . Wallach Symmetry , Representations , and Invariants Graduate Texts in Mathematics 255 Springer
In this appendix we shall find all the irreducible representations of the symmetry groups of the Platonic solids, by a mixture of geometric methods and algebraic methods similar to those used in Chapters 5 for representations of the classical groups. We shall also see how these representations occur naturally in the harmonic analysis of functions on the Platonic solids. This is a discrete analo...
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Finite tight frames have many applications and some interesting physical interpretations. One of the important subjects in this area is the ways for constructing such frames. In this article we give a concrete method for constructing finite normalized frames using Platonic solids.
متن کاملFaces of Platonic solids in all dimensions.
This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, ...
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تاریخ انتشار 2008