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Tiling With Polyominoes, Polycubes, And Rectangles, Michael Saxton
Tiling With Polyominoes, Polycubes, And Rectangles, Michael Saxton
Electronic Theses and Dissertations
In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes.
Tiling Properties Of Spectra Of Measures, John Haussermann
Tiling Properties Of Spectra Of Measures, John Haussermann
Electronic Theses and Dissertations
We investigate tiling properties of spectra of measures, i.e., sets Λ in R such that {e 2πiλx : λ ∈ Λ} forms an orthogonal basis in L 2 (µ), where µ is some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprizing tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard …
Tiling The Integers, Shasha Li
Tiling The Integers, Shasha Li
Electronic Theses and Dissertations
A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on finite Abelian groups show that Fuglede conjecture is false in high dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the finite groups ZN. Based on the Coven- Meyerowitz conjecture, it has been proved that if any spectral set in Z satisfies the the …