Tilings of convex polygons

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 -