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# Gorgeous geometry

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My personal PPT compilation of info from wiki about Platonic solids and other geometric structures.

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### Gorgeous geometry

1. 1. Geometry<br />for fun<br />(click here to play Skin-on-Skin on YouTube)<br />The Platonic solids feature prominently in the philosophy of Plato for whom they are named. Plato wrote about them in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. <br />Earth was associated with the hexahedron/cube, <br />Air with the octahedron,<br />Water with the icosahedron, and <br />Fire with the tetrahedron. <br />There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. <br />The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.<br />They are 3dimensional convex regular polyhedrons, the 3D analogs of the convex regular polygons of 2D. There are PRECISELY 5. They are unique in that the faces, edges and angles are all congruent.<br />
2. 2. Platonic Solids<br />A convex polyhedron is a Platonic solid if and only if all its faces are congruent convex regular polygons, none of its faces intersect except at their edges, and the same number of faces meet at each of its vertices. Each Platonic solid can therefore be denoted by a symbol {p, q} where p = the number of sides of each face or the number of vertices of each face) and q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).The symbol {p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have: pF = 2E - qV The other relationship between these values is given by Euler's formula V – E + F = 2. This nontrivial fact can be proved in a great variety of ways (in algebraic topology it follows from the fact that the Euler characteristic of the sphere is 2). Together these three relationships completely determine V, E, and F. Note that swapping p and q interchanges F and V while leaving E unchanged.<br />
3. 3. Platonic solids are perfectly regular solids with the following conditions: all sides are equal and all angles are the same and all faces are identical. In each corner of such a solid the same number of surfaces collide. <br />Only five Platonic solids exist: tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron.The solids and their regularities were discovered by the Pythagoreans and were called originally Pythagorean solids. The Greek philosopher Plato described the solids in detail later in his book "Timaeus" and assigned the items to the Platonic conception of the world, hence today they are well-known under the name "Platonic Solids."<br />The tetrahedron is bounded by four equilateral triangles. It has the smallest volume for its surface and represents the property of dryness. It corresponds to fire.The hexahedron is bounded by six squares. The hexadedron, standing firmly on its base, corresponds to the stable earth.The octahedron is bounded by eight equilateral triangles. It rotates freely when held by two opposite vertices and corresponds to air.The dodecahedron is bounded by twelve equilateral pentagons. It corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.The icosahedron is bounded by twenty equilateral triangles. It has the largest volume for its surface area and represents the property of wetness. The icosahedron corresponds to water.<br />Platonic Solids<br />©http://www.3quarks.com/GIF-Animations/PlatonicSolids/<br />
4. 4. Tetrahedron<br />4 faces<br />4 points<br />6 edges<br />4D hyperspherical<br />4D hypercubic<br />The heat of fire feels sharp and stabbing like little tetrahedra.<br />The tetrahedron is bounded by four equilateral triangles. <br />It has the smallest volume for its surface and represents the property of dryness.<br />
5. 5. Hexahedron<br />6 faces<br />8 points<br />12 edges<br />4D hyperspherical<br />4D hypercubic<br />The highly un-spherical solid, the hexahedron (cube) represents earth. <br />“These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. <br />
6. 6. Other Views of the Cube – 3D<br />Truncated Hexahedron<br />Hexahedron and its Dual: Octahedron<br />
7. 7. Space Filling with Polyhedrons -cube<br />Cube /Hexahedron<br />Planar Space<br />Hyperbolic Space<br />
8. 8. Other Views of 4D Cube - Tesseract<br />4D Hypercube <br />A hypercube projected into 3-space as a Schlegel diagram. There are 8 cubic cells visible, one in the center, one below each of the six exterior faces, and the last one is inside-out representing the space outside the cubic boundary. <br />Also called the Tesseract.<br />
9. 9. Other Views of Cube – 5D<br />5D Hypercube : Penteract<br />A penteract is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells. <br />The name penteract is derived from combining the name tesseract (the 4-cube) with pente for five (dimensions) in Greek. It can also be called a regular deca-5-tope or decateron, being made of 10 regular facets.<br />It is a part of an infinite family of polytopes, called hypercubes. The dual of a penteract can be called a pentacross, of the infinite family of cross-polytopes.<br />Penteract dual : Pentacross <br />Penteract orthogonal views<br />
10. 10. Octahedron<br />8 faces<br />6 points<br />12 edges<br />4D hyperspherical<br />4D hypercubic<br />The Air is made of the octahedron; its minuscule components <br />are so smooth that one can barely feel it. <br />The octahedron is bounded by eight equilateral triangles. <br />It rotates freely when held by two opposite vertices and corresponds to air.<br />
11. 11. Other Objects – Octahedron 4D<br />4D Octahedron : 24-cell.<br />24 Cell <br />Orthogonal and Net views<br />
12. 12. Icosahedron<br />20 faces<br />12 points<br />30 edges<br />4D hyperspherical<br />AKA 600-cell<br />Icosahedral Prism<br />Water, the icosahedron, flows out of one's hand when picked up,<br /> as if it is made of tiny little balls<br />The icosahedron is bounded by twenty equilateral triangles. <br />It has the largest volume for its surface area and represents the property of wetness.<br />
13. 13. Space Filling with Polyhedronsicosahedron<br />Cannot fill planar space with this object<br />Planar Space<br />120-cell<br />Hyperbolic Space<br />
14. 14. Dodecahedron<br />12 faces<br />20 points<br />30 edges<br />4D hypercubic<br />4D hyperspherical<br />AKA 120-cell<br />The dodecahedron, Plato obscurely remarks,<br /> "...the god used for arranging the constellations on the whole heaven". <br />Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) <br />and postulated that the heavens were made of this element. <br />The dodecahedron is bounded by twelve equilateral pentagons.” It corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.”<br />
15. 15. Space Filling with PolyhedronsDodecahedron<br />Planar Space<br />Cannot fill planar space with this object…<br />However, was Plato on to something?<br />The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. It is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. <br />Based on analyses of the WMAP data, cosmologists during 2004-2006 focused on the Poincaré dodecahedral space (PDS), but also considered horn topologies (which are hyperbolic) to be compatible with the data.[from wiki Shape of the Universe and Poincare pages]<br />Hyperbolic Space<br />
16. 16. Wiki links to image-packed pages<br />List of Regular Polytopes<br />Regular Polytopes basic geometry<br />Convex Uniform Honeycomb <br />