Self-similarity of quasilattices in two dimensions. I. The n-gonal quasilattice

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Abstract
The self-similarity of a two-dimensional quasilattice with an n-gonal point symmetry has been investigated for even n (n>or=8). It is shown that for any n, an n-gonal quasilattice has a self-similarity characterised by a complex number tau ; on inflation the quasilattice is scaled by mod tau mod and subsequently rotated by arg tau . tau is a PV unit of the n-cyclotomic field Q( zeta ), zeta =exp(2 pi i/n); tau satisfies (i) tau and tau -1 are both algebraic integers in Q( zeta ), (ii) mod tau mod >1 and (iii) mod tau ' mod <1 for any conjugate tau ' but tau (the complex conjugate) of tau in Q( zeta ). PV units are calculated for every n-gonal quasilattice whose multiplicity, m= phi (n/2) with phi being the Eulerian function, is less than 5; m=2 for n=8, 10, 12, m=3 for n=14, 18 and m=4 for n=16, 20, 24, 30. It is found that a quasilattice has two or more independent scales of self-similarity if its multiplicity is larger than 2.

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