On the structure of multiple translational tilings by polygonal regions

Details On the structure of multiple translational tilings by polygonal regions On the structure of multiple translational tilings by polygonal regions

1999-04-14

arxiv.org/abs/math/9904065v1

Mathematics

Metric Geometry

Scientific paper

We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon $K$ tiles the plane multiply when translated at the locations $\Lambda$, where $\Lambda$ is a multiset in the plane. The pairing property of $K$ makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for $K$ to tile with a given lattice $\Lambda$ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them) we move on to prove that a large class of such polygons tiles only quasi-periodically, which for us means that $\Lambda$ must be a finite union of translated 2-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile necessarily quasi-periodically, if at all.

Affiliated with

Also associated with

No associations

Rating

On the structure of multiple translational tilings by polygonal regions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the structure of multiple translational tilings by polygonal regions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the structure of multiple translational tilings by polygonal regions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-389135