Tiling Gap Distributions and Pair Correlation Project

Our project provides the results of numerical computations of the empirical distributions for the pair correlation, tiling point angles, and corresponding gap distributions of several well-known tilings of the plane. In particular, given the lattice points of a tiling, \(\mathbb{L}_n\), we use computations of numerical histogram plots to study the empirical statistical distributions corresponding to the (squared) pair correlation between points defined by $$(x_1-x_2)^2+(y_1-y_2)^2 \text{ for } x_i, y_i \in \mathbb{L}_n,$$ and the gap distributions $$\vartheta_{i+1} - \vartheta_{i} \text{ for } 1 \leq i < |\mathbb{L}_n| = N(R),$$ and $$s_{i+1}-s_i \text{ for } 1 \leq i < |\mathbb{L}_n|,$$ where \(\vartheta_1 < \vartheta_2 < \cdots < \vartheta_{N(R)}\) denote the angles of the tiling points and \(s_i := y_i / x_i\) denote the ordered slopes of the tiling points.

The project looks at the distributions of these statistics for both a fixed size tile that recurses inwards by repeated inflation / deflation / substitution operations, and also for a fixed dimension tiling that yields vertex points within some large radius R (e.g., the Pentagon tilings in the results below). We surmise that these two methods are equivalent.

Running the Program

The source code for the project is released under and open source license for non-commercial purposes. The program we provide contains implementations of many different tilings. The implementations of the tilings we provide in the program are easily extended in the python source code we provide with the program. Researchers and experimental mathematicians looking to compute other plots or numerically experiment with the tiling vertices we provide here are easily able to interface to our program in Sage and python. Please contact Maxie or Jayadev if you implement any new tiling variations so we can include that source within our program.

The program is run on the command line using the Sage command, or within the SageMathCloud terminal application as
sage -python tiling_pc_plots.py -t <TILING-NAME> -n <NUM_STEPS> [options],
where the optional command options are given by -q to compute pair correlation plots of the squared Euclidean distances between points, -a to compute angles of the tiling points, -d to compute the sorted angle gaps, -l to compute the tiling point slope histograms, and -g to compute the plots of the sorted slope gaps between tiling points. Alternately, a set of summary pdfs for a given tiling in the table below is generated by running the included script
$ ./generate-tiling-pdfs.sh <TILING-NAME> <NLOWER> <NUPPER>
on the Sage Cloud terminal or from a bash shell on Linux (requires pdfjam).

Results

Comparisons of the Tiling Results

The numerical computations of the empirical distributions for these tilings we present in our findings above, and in our paper, only scratch the surface of comparisons that can be made between the underlying theoretical distributions for these (and related classes of) tilings. We encourage experiemental mathematicians to compare the results of our statistical and gap distributions plots using the form below. A few classes of related tilings implemented by our software are suggested as starting points:

• Chair-Like Tilings: AmmannA4, AmmannChair, AmmannChair2, Armchair, Chair3, PChairs, Pentomino, WaltonChair
• Pentagonal Tilings: Pentagon1, Pentagon2, Pentagon3, Pentagon4, Pentagon5, Pentagon6, Pentagon10, Pentagon11, Pentagon15
• Rectangular / Domino-Shaped Tilings: Domino, Domino-9Tile, SDHouse, Squares, STPinwheel
• Triangular-Shaped Tilings: DiamondTriangle, Equithirds, GoldenTriangle, GRTriangle, Penrose, Pinwheel, SDHouse, STPinwheel, T2000Triangle, Triangles, Trihex, TriTriangle, TubingenTriangle

Links

• Tilings Encyclopedia
• Wolfram Alpha site for non-periodic substitution tilings
• Wolfram Demonstrations site for Ammann tiles
• Wolfram Demonstrations site for 15 numbered variations of pentagon tilings.
• WXML Tilings Programming HOWTO (Spring 2017) GitHub repository and Wiki site.