Distances on Rhombus Tilings

The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by the study of a quasicrystal growth model, we are here interested in better understanding how "tight" rhombus tiling spaces are flip-connected. We introduce a lower bound (Hamming-distance) on the minimal number of flips to link two tilings (flip-distance), and we investigate whether it is sharp. The answer depends on the number n of different edge directions in the tiling: positive for n=3 (dimer tilings) or n=4 (octogonal tilings), but possibly negative for n=5 (decagonal tilings) or greater values of n. A standard proof is provided for the n=3 and n=4 cases, while the complexity of the n=5 case led to a computer-assisted proof (whose main result can however be easily checked by hand).Comment: 18 pages, 9 figures, submitted to Theoretical Computer Science (special issue of DGCI'09

On substitution tilings of the plane with n-fold rotational symmetry

A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with 11-fold symmetry appearing in the literature.Comment: 17 pages, 10 figure

Distances on Lozenge Tilings

International audienceIn this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the flipdistance (i.e. the number of necessary local transformations involving three lozenges) between two given tilings. It is here proven that, for n5, We show that there is some deficient pairs of tilings for which the flip connection needs more flips than the combinatorial lower bound indicates

Principal minors and rhombus tilings

The algebraic relations between the principal minors of an $n\times n$ matrix are somewhat mysterious, see e.g. [lin-sturmfels]. We show, however, that by adding in certain \emph{almost} principal minors, the relations are generated by a single relation, the so-called hexahedron relation, which is a composition of six cluster mutations. We give in particular a Laurent-polynomial parameterization of the space of $n\times n$ matrices, whose parameters consist of certain principal and almost principal minors. The parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex $2n$-gon. A matrix is associated to an equivalence class of tilings, all related to each other by Yang-Baxter-like transformations. By specializing the initial data we can similarly parametrize the space of Hermitian symmetric matrices over $\mathbb R, \mathbb C$ or $\mathbb H$ the quaternions. Moreover by further specialization we can parametrize the space of \emph{positive definite} matrices over these rings

Structure of spaces of rhombus tilings in the lexicograhic case

Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. The zonotope $Z$ is the set $\{ Σα _iv_i 0 ≤α _i ≤m_i \}$. Each prototile used in a tiling of $Z$ is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of $Z$ is a graded poset, with minimal and maximal element