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Hitomezashi patterns, which originate from traditional Japanese embroidery,are intricate arrangements of unit-length line segments called stitches. Thestitches connect to form hitomezashi strands and hitomezashi loops, whichdivide the plane into regions. We investigate the deeper mathematicalproperties of these patterns, which also feature prominently in the study ofcorner percolation. It was previously known that every loop in a hitomezashipattern has odd width and odd height. We additionally prove that such a loophas length congruent to $4$ modulo $8$ and area congruent to $1$ modulo $4$.Although these results are simple to state, their proofs require us tounderstand the delicate topological and combinatorial properties of slicingoperations that can be applied to hitomezashi patterns. We also show that theexpected number of regions in a random $m\times n$ hitomezashi pattern (chosenaccording to a natural random model) is asymptotically$\left(\frac{\pi^2-9}{12}+o(1)\right)mn$.

Loops and Regions in Hitomezashi Patterns