Tiling by rectangles and alternating current

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.

In English and in Russian; 21 pages; 6 figures; minor improvement of exposition, Russian translation added

References in corpus (1)

• Dissections of a metal rectangle

Cited by in corpus (4)

• The boundary value problem for discrete analytic functions
• Dissection of a rectangle into rectangles with given side ratios
• Characterization of the response maps of alternating-current networks
• Superport networks