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Platonic Solids

19 18 around the globe in elegant ways Platos cosmology constructs the Elemental Solids from two types of righttriangular atom. The first atom is half an equilateral triangle, six of which then compound to produce larger equilateral triangles these go on to form the tetrahedron, octahedron and icosahedron. The second triangular atom is a diagonally halved square, which appears in fours, making squares which then form cubes. The Platonic Solids have planes of symmetry dividing them into mirror image halves, the tetrahedron has six, the octahedron and cube have nine and the icosahedron and dodecahedron have fifteen. When the tetrahedron, octahedron and icosahedron are constructed from Platos triangular atoms, paths are defined which make their mirror planes explicit. The cube however needs twice as many triangular divisions as Plato gave it top row to delineate all its mirror planes middle row. Projecting the subdivided Platonic Solids onto their circumspheres produces three spherical systems of symmetry. Each spherical system is defined by a characteristic spherical triangle with one right angle, and one angle of one third of a half turn. Their third angles are respectively one third of a half turn top row, one quarter of a half turn middle row and one fifth of a half turn lower row. This sequence of , , and elegantly inverts the Pythagorean whole number triple 3, 4, 5. 1 3 14 15
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