What is the minimum perimeter for a tile on the unit square lattice? A unit square has perimeter 4, but this is not minimal. In 1989, Jaigyoung Choe determined the minimal perimeter to be [1]. In this Demonstration, the Choe irregular hexagons are shown in blue (or orange). It turns out that this tile was already well known for hexagons within an optimized Cairo tessellation.

[1] J. Choe, "On the Existence and Regularity of Fundamental Domains with Least Boundary Area," Journal of Differential Geometry, 29(3), 1989 pp. 623–663. 10.4310/jdg/1214443065.