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History The five Platonic solids have been known since antiquity. The neolithic people of Scotland constructed stone models of all five solids at least 1000 years before Plato (Atiyah and Sutcliffe 2003). Plato, for whom they are named, 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. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. Regarding the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the gods used for arranging the constellations on the whole heaven". See: http://en.wikipedia.org/wiki/Platonic_solid The duals of Platonic solids are other Platonic solids and, in fact, the dual of the tetrahedron is another tetrahedron. Properties of Platonic solids 1. The vertices of all Platonic solids lie on a sphere. 2. All the dihedral angles are equal. 3. All the vertex figures are regular polygons. 4. All the solid angles are equivalent. 5. All the vertices are surrounded by the same number of faces.
Demonstration Large circumspheres of the Platonic Regular Solids are a visual/tactile/kinetic teaching tool for Open Day on Friday, 8th September 2006. A 1200 x 2400mm sheet of polycarbonate was cut into 70 lengths of 30 x 1200 mm with a hole in each end. Twelve 3.5mm steel bolts will each tie 5 edges to outline the circumsphere of the icosahedron. The Archimedean semi-regular polyhedral dome is constructed from 35 edges into a half icosidodecahedron. The icosahedral frame is rigid so the edges can be free move around the bolt. The icosahedron is the Platonic solid having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces. Every polyhedron has a dual, another polyhedron in which faces and polyhedron vertices occupy complementary locations. The dodecahedron is the dual of the icosahedron. It has the same number of edges (30) and each of the twelve faces corresponds to the icosohedral vertices. The corresponding 20 vertices of the dodecahedral frame must be locked into position or the structure will twist at the vertices and collapse. Similarly, the cubic frame will collapse with mobile vertices while the dual octahedral frame is rigid. This is one reason conventional buildings suffer such damage in storms. We prefer orthogonal buildings because of maximal horizontal floor space and the efficient intersection of horizontal walls. The floors of the world trade centre collapsed catastrophically like a deck of cards on September 11, 2001. A non-orthogonal structure such as honeycomb is stronger and would have suffered less damage. Minimal Surfaces A sphere is strong under uniform pressure. In nature, hedgehogs and virulent viruses form spheres for defense. Observe bubbles on the surface of soapy water. They form spheres intersecting as circular arches or arcs. The sphere has maximal volume for a given surface area or minimal surface area for a given volume. This property of minimal surface area applies to soap films suspended about any boundary frame. For example, the catenoid curve is a surface of revolution of the trigonometric cosh() function and can be demonstrated by the suspended film between 2 facing circles. As well as minimal area, the catenoid has zero mean curvature. This means there are no concave or convex points on the surface. Every point is a saddle point. This gives the catenoid the same neutral aerodynamics as a plane flat surface. There is no differential Bernoulli effect, no valleys or grooves causing wind-tunnel effects such as the shaping of sand dunes or the flapping of a loose sails at sea. A large catenoid canopy is lightweight and stable in high winds. A solid catenoid is stronger than a flat or convex surface. The minimal surface property means economy of materials and the economy of scale. Large structures can be built if they are the right shape. Applications
Further work We need further research into tough, UV resistant polymers such as Polybutylene terephthalate or Polyetherimide. |