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The Platonic Solids are so important to geometry that I thought to make them as affordable as possible. All 5 are the same approx height.
The best way to understand these geometries is as symmetrical packing of spheres. If you have 4 identical spheres and squash them together, their centers will naturally arrange themselves in the shape of a tetrahedron, the simplest straight edged 3 dimensional form. Note that the distances and angles between all vertexes is identical.
Perhaps a little less obvious (and less stable), 6 spheres will form an Octahedron.
The more spheres you add the less stable it becomes, but if you add the proviso that all the angles and distances must remain identical then the Cube, the Icosahedron, and the Dodecahedron, are the only other possible ways for the spheres to cluster.
Therefore these 5 shapes are known as the Perfect solids, or the Regular Convex Polyhedrons. They and their modifications are the primary building blocks, the path of energy efficiency for stable 3 dimensional forms from molecules to large scale distribution of galaxies.
Two tetrahedrons interlocking into a 'star tetrahedron' form the vetexes of a cube externally and an octahedron internally. These 3 forms are the 'cuboctet' family of shapes. They have equivalents in all dimensions known as the Simplex (tetrahedral: minimum points to contain space), the Measure (cubic: volume of space), and the Cross (octahedral: equal extension).
The other two Platonics, the icosahedron and the dodecahedron, are the 5 symmetry group forms. They are 'duals' of each other the same way the cube and octahedron are. These forms are far less stable. They embody the Golden Mean Ratio and tend to appear as the creation of living beings, whereas the cuboctet family appear in all matter and energy. Interesting then that Scientific American had a cover article announcing that the Universe appears to be dodecahedral!
Much more info about these enchanting forms can be found at SacredGeometryWeb.com