Element number of the Platonic solids |
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| Authors: | J Akiyama H Maehara G Nakamura I Sato |
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| Institution: | (1) Materials Structure and Modeling Research Group of the Hungarian Academy of Sciences at Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary;(2) Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary;; |
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| Abstract: | Let Σ be a set of polyhedra. A set Ω of polyhedra is said to be an element set for Σ if each polyhedron in Σ is the union
of a finite number of polyhedra in Ω. We call each polyhedron of the element set Ω an element for Σ. In this paper, we determine
one element set for the set Π of the Platonic solids, and prove that this element set is, in fact, best possible; it achieves
the minimum in terms of cardinality among all the element sets for Π. We also introduce the notion of indecomposability of
a polyhedron and present a conjecture in Sect. 3. |
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