Pair Correlation and Gap Distributions for Substitution Tilings and Generalized Ulam Sets in the Plane

TL;DR

This paper analyzes statistical and gap distributions of various plane tilings and generalized Ulam sets, providing empirical data, theoretical proofs, and open-source software for further exploration.

Contribution

It introduces new empirical analyses and theoretical results for tilings and Ulam sets, along with open-source software for distribution approximation.

Findings

Distribution patterns for Ammann Chair tiling identified

Asymptotic formula for Ulam set entry times proved

Software enables exploration of various tilings and point sets

Abstract

We study empirical statistical and gap distributions of several important tilings of the plane. In particular, we consider the slope distributions, the angle distributions, pair correlation, squared-distance pair correlation, angle gap distributions, and slope gap distributions for the Ammann Chair tiling, the recently discovered fifteenth pentagonal tiling, and a few pertinent tilings related to these famous examples. We also consider the spatial statistics of generalized Ulam sets in two dimensions. Additionally, we carefully prove a tight asymptotic formula for the time steps in which Ulam set points at certain prescribed geometric positions in their plots in the plane formally enter the recursively-defined sets. The software we have developed to these generate numerical approximations to the distributions for the tilings we consider here is written in Python under the Sage environment and is released as open-source software which is available freely on our websites. In addition to the small subset of tilings and other point sets in the plane we study within the article, our program supports many other tiling variants and is easily extended for researchers to explore related tilings and iterative sets.