Abstract
This article provides an introduction to techniques developed to address three types of tiling questions in the integer lattice: questions of tileability - which regions in a collection\(\mathcal{R}\) can be tiled by a tile set T; connectivity - what can be said about how two tilings of a region in\(\mathcal{R}\) must be related; and enumeration - how many ways can a region in\(\mathcal{R}\) be tiled by T. We place an emphasis on tile invariants, linear combinations among the number of copies of each tile that must persist in any tiling of a region. A tiler’s toolbox draws on content from combinatorics, number theory, group theory, and topology. Given their hands-on nature, their variety, and their relevance to the undergraduate curriculum, tiling questions continue to be a lively subject for undergraduate mathematics research.
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Hitchman, M.P. (2017). Tile Invariants for Tackling Tiling Questions. In: Wootton, A., Peterson, V., Lee, C. (eds) A Primer for Undergraduate Research. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66065-3_3
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DOI: https://doi.org/10.1007/978-3-319-66065-3_3
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