Families of convex tilings


  • Richard Kenyon Department of Mathematics, Yale University, New Haven, CT 06520, USA




 We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile "orientations" determine the tiling. We associate to a tiling an underlying bipartite planar graph $\mathcal{G}$ and its corresponding Kasteleyn matrix $K$. If $\mathcal{G}$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.


Download data is not yet available.


A. Abrams and R. Kenyon, Fixed-energy harmonic functions, Discrete Anal. 2017, Paper No. 18, 21 pp. MR 3734203.

M. Dehn, Über Zerlegung von Rechtecken in Rechtecke, Math. Ann. 57 (1903), no. 3, 314–332. MR 1511212.

A. B. Goncharov and R. Kenyon, Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 5, 747–813. MR 3185352.

P. W. Kasteleyn, Graph theory and crystal physics, in Graph Theory and Theoretical Physics, 43–110, Academic Press, London, 1967. MR 0253689.

R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 5, 591–618. MR 1473567.

R. Kenyon, Lectures on dimers, in Statistical Mechanics, 191–230, IAS/Park City Math. Ser., 16, Amer. Math. Soc., Providence, RI, 2009. MR 2523460.

R. Kenyon, Tilings and discrete Dirichlet problems, Israel J. Math. 105 (1998), 61–84. MR 1639727.

R. W. Kenyon and S. Sheffield, Dimers, tilings and trees, J. Combin. Theory Ser. B 92 (2004), no. 2, 295–317. MR 2099145.

A. Postnikov, Total positivity, Grassmannians, and networks, https://arxiv.org/abs/math/0609764 [math.CO], 2006.

O. Schramm, Existence and uniqueness of packings with specified combinatorics, Israel J. Math. 73 (1991), no. 3, 321–341. MR 1135221.

W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, in The Epstein Birthday Schrift, 511–549, Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, 1998. MR 1668340.

W. T. Tutte, Dissections into equilateral triangles, in The Mathematical Gardner, 127–139, Prindle, Weber & Schmidt, Boston, 1981. MR 593154.

S. Wimer, I. Koren and I. Cederbaum, Floorplans, planar graphs, and layouts, IEEE Trans. Circuits and Systems 35 (1988), no. 3, 267–278. MR 0931855.