# The density and minimal gap of visible points in some planar quasicrystals

## TL;DR

This paper derives formulas for the density of visible points in certain planar quasicrystals and uses these to compute the minimal gaps between angles of visible points, revealing structural properties of these aperiodic sets.

## Contribution

It provides explicit formulas for the density of visible points and calculates the minimal angular gaps in specific planar quasicrystals, advancing understanding of their geometric distribution.

## Key findings

- Formulas for the density of visible points in quasicrystals
- Calculation of minimal normalized gaps between visible point angles
- Application to Ammann-Beenker and Penrose tilings

## Abstract

We give formulas for the density of visible points of several families of planar quasicrystals, which include the Ammann-Beenker point set and vertex sets of some rhombic Penrose tilings. These densities are used in order to calculate the limiting minimal normalised gap between the angles of visible points in two families of planar quasicrystals, which include the Ammann-Beenker point set and vertex sets of some rhombic Penrose tilings.