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The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question—one that requires an explicit construction.
This book explains Kepler's cosmological theory, based on the Copernican system, in which the five Platonic solids dictate the structure of the universe and reflect God's plan through geometry. This was virtually the first attempt since Copernicus to say that the theory of heliocentrism is physically true. [2]
The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic (χ) of 2.
The five Platonic solids have an Euler characteristic of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of any polyhedron which is star-shaped with respect to some interior point.
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 Fifty-Nine Icosahedra. [58]
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 five Platonic solids were known to them. Pythagoras knew of at least three of them and Theaetetus (c. 417 BC – 369 BC) described all five. Later, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory.
The regular icosahedron, as one of the Platonic solids, is a regular polyhedron. It is isogonal , isohedral , and isotoxal : any two vertices, two faces, and two edges of a regular icosahedron respectively can be transformed by rotations and reflections under its symmetry orbit, which preserves the appearance.