Tiling by rectangles and alternating current

Details Tiling by rectangles and alternating current Tiling by rectangles and alternating current

2010-02-06

arxiv.org/abs/1002.1356v1

J. Combinatorial Theory, Series A 118:3 (2011) 920-937

Mathematics

Combinatorics

19 pages; 4 figures

Scientific paper

10.1016/j.jcta.2010.11.012

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of the paper is an application of alternating-current circuits. The following results are obtained: - a necessary condition for a rectangle to be tilable by rectangles of given shapes; - a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it; - a criterion for a generic polygon to be tilable by squares. These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne and G. Szekeres.

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