SIMPLE GEOMETRICAL INTERPRETATIONS OF QUASI-PERIODIC POINT SETS BY VARIOUS CONCEPTS OF INTERPENETRATION

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Abstract
Behind several methods to describe quasicrystals there is some concept of interpenetration. Two such concepts are outlined in this paper, giving a geometrically attractive interpretation of quasi-periodic (QP) point sets. First we show how QP tilings constructed with the grid method can also be obtained by taking the mass centre of suitably chosen sets of points belonging to a union of periodic lattices (intertwinning). In a second part of the paper it is shown how from the projection method QP point sets can be obtained that can be decomposed into a superposition of distorted lattices (intergrowth1). These decompositions are derived for the case where a face-centered hypercubic Bravais lattice generates the pattern. A general study of such decompositions may be of heuristic value for structure determinations.

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