Approximate indexing of icosahedral polyhedra with fibonacci numbers

Abstract

In general, indexing faces of icosahedral face-forms requires irrational numbers. However, for many practical purposes an approximate indexing based on triplets of integer numbers can be used. Two possible approaches called, respectively, “Fibonacci Matrix Methods” (FMM) and the “Linear Combination Method” (LCM) are described. FMM relies on the use of “auxiliary” matrices F_n, F² _n, F³ _n and F⁴ _n which have Fibonacci numbers as their elements. These matrices allow good approximation of the results usually obtained using the standard five-fold rotation matrices which are typical of icosahedral symmetry. LCM is based on the use of a classical crystallographic rule i.e. the so-called “Goldschmidt Complication Law” which is just a particular case of linear combination of triplets of face indices, with integers as coefficients. The occurrence of large integer indices is remarked.

Key Words:

• Icosahedral symmetry
• indexing polyhedra
• quasicrystals