Choe's Hexagon and Cairo Tiling

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.


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[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.
[2] J. Cepelewicz, "Mathematicians Complete Quest to Build 'Spherical Cubes'," Quanta Magazine (Feb 28, 2023).
[3] E. Pegg. "Choe's Irregular Hexagon" from Wolfram Community–A Wolfram Web Resource.
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