Title: Pair Correlation and Gap Distributions for Substitution Tilings and Generalized Ulam Sets in the Plane,
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.