Tiling the integers with aperiodic tiles

Abstract

Abstract.A finite subset A of integers tiles the discrete line Z if the integers can be written as a disjoint union of translates of A. In some cases, necessary and sufficient conditions for A to tile the integers are known. We extend this result to a large class of nonperiodic tilings and give a new formulation of the Coven-Meyerowitz reciprocity conjecture which is equivalent to the Flugede conjecture in one dimension.

Music in the pedagogy of mathematics

“…The subject is a natural candidate for the development of didactic materials, given the quantity and quality of mathematics involved that is not usually studied in standard courses, such as periodic sets, Hajós and non-Hajós groups, Fourier analysis on finite groups, etc. See Vuza (1991Vuza ( , 1992aVuza ( , 1992bVuza ( , 1993, Agon (2009), Amiot (2009), Kolountzakis andMatolcsi (2009), andJedrzejewski (2009). Together with these subjects we find a variety of musical examples that illustrate the mathematical theory and that can be understood without much need of traditional music theory notation.…”

Quasiperiodic music

“…These rhythmic structures arise from combining aperiodic canons to create another canon having a large period. These canons are closely related to the problem of tiling the integers with aperiodic tiles (as in the title of Jedrzejewski, 2009). Again, there is an interplay between periodicity and aperiodicity, and complex patterns arising from repetitions of simpler ones.…”

Enumeration of Vuza Canons

“…In 2004 at the Mamux Seminar, F. Jedrzejewski gave a solution for constructing large Vuza canons, depending only on the parameters n 1 , n 2 , n 3 , p 1 and p 2 (see (Jedrzejewski , 2006(Jedrzejewski , , 2009). Let us rephrase this result in the following manner: Theorem 12.…”