Tiling by rectangles and alternating current

TL;DR

This paper explores tilings of polygons by rectangles using alternating-current circuit models, providing new criteria for tilability and generalizing previous results in the field.

Contribution

It introduces an innovative approach applying AC circuits to tiling problems, deriving new necessary and sufficient conditions for tilings by rectangles and squares.

Findings

A necessary condition for a rectangle to be tiled by given shapes.

A criterion for tiling a rectangle with similar but not homothetic rectangles.

A criterion for tiling a generic polygon with squares.

Abstract

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.