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In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
The graphs of the Platonic solids have been called Platonic graphs. As well as having all the other properties of polyhedral graphs, these are symmetric graphs, and all of them have Hamiltonian cycles. [9] There are five of these graphs: Tetrahedral graph – 4 vertices, 6 edges; Octahedral graph – 6 vertices, 12 edges
Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra; the shapes of these creatures are indicated by their names. [5] The outer protein shells of many viruses form regular polyhedra. For example, HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus. [6] [7]
The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube . Table of polyhedra
The polytopes of rank 2 (2-polytopes) are called polygons.Regular polygons are equilateral and cyclic.A p-gonal regular polygon is represented by Schläfli symbol {p}.. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular.
Able to conform to the shape of its container but retains a (nearly) constant volume independent of pressure. Gas: A compressible fluid. Not only will a gas take the shape of its container but it will also expand to fill the container. Mesomorphic states: States of matter intermediate between solid and liquid.
Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction) Coxeter The Beauty of Geometry: Twelve Essays , Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 3: Wythoff's Construction for Uniform Polytopes)