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We study empirical statistical and gap distributions of several importanttilings of the plane. In particular, we consider the slope distributions, theangle distributions, pair correlation, squared-distance pair correlation, anglegap distributions, and slope gap distributions for the Ammann Chair tiling, therecently discovered fifteenth pentagonal tiling, and a few pertinent tilingsrelated to these famous examples. We also consider the spatial statistics ofgeneralized Ulam sets in two dimensions. Additionally, we carefully prove atight asymptotic formula for the time steps in which Ulam set points at certainprescribed geometric positions in their plots in the plane formally enter therecursively-defined sets.  The software we have developed to these generate numerical approximations tothe distributions for the tilings we consider here is written in Python underthe Sage environment and is released as open-source software which is availablefreely on our websites. In addition to the small subset of tilings and otherpoint sets in the plane we study within the article, our program supports manyother tiling variants and is easily extended for researchers to explore relatedtilings and iterative sets.

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