# GRID 2020 Abstracts

## Locating Facilities by the Properties of the Hexagonal Regular Grid

Dr. Shai Gul
Faculty of Sciences, HIT & Mathematics Department, Bar-Ilan University

Together with:
Prof. Reuven Cohen

Session IV: Grids of Biology and Technology
Monday, November 23, 2020 | 14:30-15:30 (Duet B)

We will give a theoretical approach how to locate international airports in the US such that for every citizen will have a minimal distance to the nearest airport. The grid, which leads to the desired minimization, is the hexagonal regular grid. In civil engineering, the right grid can reduce extremely the "costs" in locating facilities.

In the theoretical approach, the problem is to cover a domain by unit discs (each center of disc represents an airport). W. Blaschke, determined an upper bound to the number of unit discs which are needed to cover a given convex domain Ω. Blaschke showed that a domain can be covered with

$${ [ { \frac{2}{3\sqrt{3}} A + \frac{2}{\pi\sqrt{3}} L + 1 } ] }$$

, unit circles, where A is the area of the given domain and L the perimeter. This result is due to the properties of the hexagonal grid. This talk will be composed of three main results. L. Fejes T'oth showed that the hexagonal regular grid is the optimal grid. In this work: First, we will show that in specific cases Blaschke’s result can be improved, we will show how to locate the hexagonal grid in these cases. Second, we will give a sufficient condition under which Blaschke's expression can be improved. Third, we will give an algorithmic approach, which determines the exact position of the hexagonal grid, such that the number of unit hexagons (in the hexagonal lattice) which hit Ω is minimized. This result determines us the exact location of airports in the US.

This manuscript is accessible to civil engineering's, computer scientists, designers and mathematicians.