Abstract
We show that every tiling of a convex set in the Euclidean plane \(\mathbb {R}^2\) by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane \(\mathbb {R}^2\), which is based on a surprising connection to a random walk on a directed graph.
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M. W. was supported by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes) and the German Research Foundation (DFG) via RTG 1523.
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Richter, C., Wirth, M. Tilings of Convex Sets by Mutually Incongruent Equilateral Triangles Contain Arbitrarily Small Tiles. Discrete Comput Geom 63, 169–181 (2020). https://doi.org/10.1007/s00454-019-00061-6
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DOI: https://doi.org/10.1007/s00454-019-00061-6