Affine symmetry groups in 2D-quasicrystals. (Q2519171)

Let \(Q\) be a ``cut and project'' model of a quasicrystal embedded into a hyperspace \(E=U\oplus V\) where \(U\) is a physical space and \(V\) is a phase space. It is assumed that there is a lattice \(M\) in \(E\) with a base \(e_1,\dots,e_n\). A window \(W\) in \(V\) is the projection of the unit cube in \(E\) constructed on \(e_1,\dots,e_n\). Under these assumptions it is shown that \(W\) is a polygon with even number of sides which are parallel in pairs. This fact is used for a classification of finite symmetry groups of \(Q\) which consist of affine transformations of \(E\) preserving \(U\) and \(M\).