Perimeter-minimizing pentagonal tilings (Q2511309)

A pentagonal tiling of the Euclidean plane is a cover of the plane by non-overlapping pentagons, which are called tiles. The perimeter ratio of a tiling is the limit superior as \(r\) tends to infinity of the sum of perimeters of all tiles in a disc of radius \(r\) divided by the area of the disc. One focus of the paper is on tilings by pentagons of unit area that realize the smallest possible perimeter ratio. The investigations include monohedral tilings, where all tiles are isometric images of each other, tilings which are invariant under two linearly independent translations, and hence give rise to tilings of flat tori, but also tilings without translational symmetry. A particular emphasis is on perimeter minimizing tilings of flat tori by pentagons (or hexagons) of unit area.