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The 13 Archimedean Solids have all their edges the same length and all their corners have exactly the same variety of shapes meeting. These 13 are the only possible such solids in 3 dimensions. They combine the symmetries of the platonic solids, and most of them show simple relationships to the platonic solids as they can be made by truncating (cutting off the corners or edges) or combining the platonics and joining their vertexes.
The first historical reference to these 13 shapes is in a book by the Greek mathematician Pappus of Alexandria who lived around 300AD. He attributes them to Archimedes although we have no existing text of Archimedes mentioning them. “Although many solid figures having all kinds of surfaces can be conceived, those which appear to be regularly formed are most deserving of attention. Those include not only the five figures found in the godlike Plato, that is, the tetrahedron and the cube, the octahedron and the dodecahedron, and fifthly the icosahedron, but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons. ”
The importance of these shapes and the platonics to Sacred Geometry Science lies in their being the lowest entropy states for atoms to arrange themselves in. These are the most balanced distributions of points around a central point. All other things being equal (which of course they never are) energetic system will naturally tend to be more stable when assuming these states.
All green geometries on this page are from http://commons.wikimedia.org/wiki/Special:Contributions/Hellisp