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We establish a historical context for the Platonic solids, show various properties of their features, and prove why there can be no more than ve in total. We will also discuss the nite groups of symmetries on a line, in a plane, and in three dimensional space.
Timaeus, the regular polyhedrons came to be known as the Platonic solids. The beauty and fascinating mathematical properties of these five forms haunted scholars from the time of Plato through the Renaissance. The analysis of the Platonic solids provides the cli-mactic final book of Euclid’s Elements. Johannes Kepler believed
This article will discuss the group symmetries of the Platonic solids using a variety of concepts, including rotations, re°ections, permutations, matrix groups, duality, homomorphisms, and representations.
The name Platonic solid refers to their prominent mention in Plato’s Timaeus, one of his most speculative dialogues, in which Plato posited that each of the four classical elements is made up of one of the regular polyhedra. Fire is composed of tetrahedra; Earth is composed of cubes; Air is made up of octahedra; Water is made up of icosahedra.
A Platonic solid is a convex polyhedron whose faces are all congruent regular polygons, with the same number of faces meeting at each vertex. In some sense, these are the most regular and most symmetric polyhedra that you can find. Our goal now will be to classify the Platonic solids — in other words, hunt them all down.
The five Platonic solids icosahedron Quintessence cube Water octahedron Air dodecahedron . Created Date: 2/19/2011 9:13:27 AM ...
31 Δεκ 2016 · In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria: Geometers have studied the mathematical beauty and symmetry of the Platonic solids for thousands of years.[1] .
