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