Weighted distances in the Cairo pattern

Abstract

The Cairo Pattern is an interesting tiling of the plane. It consists of pentagons which completely cover the plane without overlapping. The Cairo Pattern is the dual of a semi-regular grid. It is assumed that one can walk on the pentagons such that it is possible to step only two neighboring pentagons. Different types of steps have different weights. The distance of two pentagons is the length of the minimal path between them. All distances are based on a linear programming model. The main tool of the analysis is the simplex method of linear programming. The analysis also uses the special properties of the model of minimal path. All minimal paths are given explicitly. Therefore, the complexity of our final result is O(1) inspite of the fact that the general simplex method is thought to be exponential.