Tilings of convex polygons

Richard Kenyon

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 3, page 929-944
  • ISSN: 0373-0956

Abstract

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Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon P a quadratic form q ( P ) , which must be positive semidefinite if P is tileable with rational polygons.The above results also hold replacing the rationality condition with the following: a polygon P is coordinate-rational if a homothetic copy of P has vertices with rational coordinates in 2 .Using the above results, we show that a convex polygon P with angles multiples of π / n and an edge from 0 to 1 can be tiled with triangles having angles multiples of π / n if and only if vertices of P are in the field [ e 2 π i / n ] .

How to cite

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Kenyon, Richard. "Tilings of convex polygons." Annales de l'institut Fourier 47.3 (1997): 929-944. <http://eudml.org/doc/75250>.

@article{Kenyon1997,
abstract = {Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon $P$ a quadratic form $q(P)$, which must be positive semidefinite if $P$ is tileable with rational polygons.The above results also hold replacing the rationality condition with the following: a polygon $P$ is coordinate-rational if a homothetic copy of $P$ has vertices with rational coordinates in $\{\Bbb R\}^2$.Using the above results, we show that a convex polygon $P\in \{\Bbb C\}$ with angles multiples of $\pi /n$ and an edge from $0$ to $1$ can be tiled with triangles having angles multiples of $\pi /n$ if and only if vertices of $P$ are in the field $\{\Bbb Q\}[e^\{2\pi i/n\}]$.},
author = {Kenyon, Richard},
journal = {Annales de l'institut Fourier},
keywords = {tilings; quadratic forms; convexity},
language = {eng},
number = {3},
pages = {929-944},
publisher = {Association des Annales de l'Institut Fourier},
title = {Tilings of convex polygons},
url = {http://eudml.org/doc/75250},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Kenyon, Richard
TI - Tilings of convex polygons
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 3
SP - 929
EP - 944
AB - Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon $P$ a quadratic form $q(P)$, which must be positive semidefinite if $P$ is tileable with rational polygons.The above results also hold replacing the rationality condition with the following: a polygon $P$ is coordinate-rational if a homothetic copy of $P$ has vertices with rational coordinates in ${\Bbb R}^2$.Using the above results, we show that a convex polygon $P\in {\Bbb C}$ with angles multiples of $\pi /n$ and an edge from $0$ to $1$ can be tiled with triangles having angles multiples of $\pi /n$ if and only if vertices of $P$ are in the field ${\Bbb Q}[e^{2\pi i/n}]$.
LA - eng
KW - tilings; quadratic forms; convexity
UR - http://eudml.org/doc/75250
ER -

References

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  1. [1] C. BAVARD, E. GHYS, Polygones du plan et polyèdres hyperboliques., Geometriae Dedicata, 43 (1992), 207-224. Zbl0758.52001MR93k:52009
  2. [2] J.H. CONWAY, J.C. LAGARIAS, Tilings with polyominoes and combinatorial group theory, J. Combin. Theory Ser. A., 53 (1990), 183-206. Zbl0741.05019MR91a:05030
  3. [3] M. LACZKOVICH, Tilings of polygons with similar triangles, Combinatorica, 10 (1990), 281-306. Zbl0721.52013MR92d:52045
  4. [4] R. KENYON, A group of paths in ℝ2, Trans. A.M.S., 348 (1996), 3155-3172. Zbl0896.52024MR96j:20054
  5. [5] W.P. THURSTON, Shapes of polyhedra, Univ. of Minnesota, Geometry Center Research Report GCG7. 
  6. [6] W.T. TUTTE, The dissection of equilateral triangles into equilateral triangles, Proc. Camb. Phil. Soc., 44 (1948), 463-482. Zbl0030.40903MR10,319c

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