Tiling by rectangles and alternating current

M. V. Prasolov; Mikhail Skopenkov

doi:10.1016/j.jcta.2010.11.012

Tiling by rectangles and alternating current

M. V. Prasolov, Mikhail Skopenkov

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings by R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of this 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 those of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne, and G. Szekeres. © 2010 Elsevier Inc.

Original language	English (US)
Pages (from-to)	920-937
Number of pages	18
Journal	Journal of Combinatorial Theory, Series A
Volume	118
Issue number	3
DOIs	https://doi.org/10.1016/j.jcta.2010.11.012
State	Published - Apr 2011

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors are grateful to A. Akopyan, M. Houston, and B. George for useful discussions. M. Skopenkov was supported in part by INTAS grant 06-1000014-6277, Russian Foundation of Basic Research grant 06-01-72551-NCNIL-a, Moebius Contest Foundation for Young Scientists and Euler Foundation.

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Theoretical Computer Science

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10.1016/j.jcta.2010.11.012

Cite this

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title = "Tiling by rectangles and alternating current",

abstract = "This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings by R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of this 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 those of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne, and G. Szekeres. {\textcopyright} 2010 Elsevier Inc.",

author = "Prasolov, {M. V.} and Mikhail Skopenkov",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The authors are grateful to A. Akopyan, M. Houston, and B. George for useful discussions. M. Skopenkov was supported in part by INTAS grant 06-1000014-6277, Russian Foundation of Basic Research grant 06-01-72551-NCNIL-a, Moebius Contest Foundation for Young Scientists and Euler Foundation.",

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doi = "10.1016/j.jcta.2010.11.012",

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AU - Prasolov, M. V.

AU - Skopenkov, Mikhail

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The authors are grateful to A. Akopyan, M. Houston, and B. George for useful discussions. M. Skopenkov was supported in part by INTAS grant 06-1000014-6277, Russian Foundation of Basic Research grant 06-01-72551-NCNIL-a, Moebius Contest Foundation for Young Scientists and Euler Foundation.

PY - 2011/4

Y1 - 2011/4

N2 - This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings by R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of this 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 those of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne, and G. Szekeres. © 2010 Elsevier Inc.

AB - This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings by R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of this 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 those of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne, and G. Szekeres. © 2010 Elsevier Inc.

UR - http://hdl.handle.net/10754/561608

UR - http://arxiv.org/abs/arXiv:1002.1356v1

UR - http://www.scopus.com/inward/record.url?scp=78751576076&partnerID=8YFLogxK

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DO - 10.1016/j.jcta.2010.11.012

M3 - Article

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SP - 920

EP - 937

JO - Journal of Combinatorial Theory, Series A

JF - Journal of Combinatorial Theory, Series A

IS - 3

ER -

Tiling by rectangles and alternating current

Abstract

Bibliographical note

ASJC Scopus subject areas

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