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Growing icosahedra. (English) Zbl 0882.52006

From the text: “I first divided an icosahedron into ten types of small modules, then reconstructed various polyhedra by combining the modules radially and symmetrically. I found a beautiful hierarchical structure in the polyhedra, where in each layer in the reconstruction process there appear various 5-fold symmetrical polyhedra including the dedocahedron, icosahedron, and rhombic triacontahedron. Furthermore, it is possible to fill the entire space in these models while maintaining a hierarchical structure in the filling pattern.
Although this discovery was done outside the territory of translational symmetry in crystallography, it is quite suggestive of a solution to the structure of 5-fold symmetrical crystals that grow radially and symmetrically in the natural world”.

MSC:

52B15 Symmetry properties of polytopes
82D25 Statistical mechanics of crystals
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