Wolfram Demonstrations Project

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.

Contributed by: Ed Pegg Jr

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References

[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). www.quantamagazine.org/mathematicians-complete-quest-to-build-spherical-cubes-20230210/#0.

[3] E. Pegg. "Choe's Irregular Hexagon" from Wolfram Community–A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/2840845.