Let a, b and c be positive integers and consider a hexagon with side lengths a,b,c,a,b,c whose angles are 120◦ (see Figure 1). The subject of our interest is rhombus tilings of such a hexagon using rhombi with all sides of length 1 and angles 60◦ and 120◦. Figure 2 shows an example of a rhombus tiling of a hexagon with a = 3, b = 5 and c = 4. A first natural question to be asked is how many rho...

We study some decision problems concerning the tiling of the plane with Wang tiles We present a proof for the undecidability of the tiling problem for the whole plane and also for the periodic tiling In these poofs an aperiodic tile set is constructed We are interested in the second part in the link between the cardinality of possible tilings of the plane with a given tile set and the existence...

In this paper I have settled the open problem, posed by J. Marshall Ash and S. Golomb in [4], of tiling an m × n rectangle with L-shaped trominoes, with the condition that 3 |(mn 2) and a domino is removed from the given rectangle. It turns out that for any given m, n ≥ 7, the only pairs of squares which prevent a tiling are {(1,2), (2,2)}, {(2,1), (2,2)}, {(2,3), (2,4)}, {(3,2), (4,2)} and the...

This work studies the application of the discrete Hölder-Brascamp-Lieb (HBL) inequalities to the design of communication optimal algorithms. In particular, it describes optimal tiling (blocking) strategies for nested loops that lack data dependencies and exhibit linear memory access patterns. We attain known lower bounds for communication costs by unraveling the relationship between the HBL lin...

we present a new renormalization-group calculation of the octagonal tiling excitation spectrum. Two new self-;imilar octagonal tilings are introduced, a triangle-Lite and a triangle-dan tiling, which are entangled with the standard rhombus-~quare tiling. The decimation procedure involved in the renormalization group calculation transforms each tiling enta a different one and goe; back tu the (r...

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), which may or may not be similar to ABC. We wish to characterize the numbers N for which some triangle ABC can be tiled by N tiles, or more generally to characterize the triples (N,T ) such that ABC can be N-tiled using tile T . In the first part of the paper we exhibit certain families of tilings whic...