Original scientific paper

On the Complexity of Platonic Solids

Danail Bonchev ; Program for the Theory of Complex Natural Systems, Texas A&M University, Galveston, TX 77551, USA

Abstract

Global, relative, and local complexity of the five Platonic solids (tetrahedron, octahedron, cube, icosahedron, and dodecahedron) are described and compared. Several of the most recent measures of topological complexity are used: the subgraph count, overall connectivity and overall Wiener indices, the total walk count, and the information theoretic index for vertex degrees distribution. Equations are derived for the first several orders of these indices as functions of the number of vertices and vertex degrees. Relative complexity, defined as the ratio of the complexity index selected and its value for the complete graph having the same number of vertices as the respective Platonic solid, singles out tetrahedron as the most complex structure with 100 % relative complexity. The global complexity indices, as well as the local indices (defined per vertex and per edge) uniformly identify icosahedron as the most complex Platonic solid. These findings correlate with the preferable formation of icosahedron and tetrahedron in a variety of cases.

Keywords

Platonic solids; complexity; subgraph count; overall connectivity; overall Wiener; total walk count

Hrčak ID:

102661

URI

https://hrcak.srce.hr/102661

Publication date:

31.5.2004.

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