Year: 2026
Type: preprint
Abstract: Hitomezashi is a particular style of sashiko stitching characterized by crossing lines of stitches. Certain hitomezashi designs can be encoded with two binary strings and have been the subject of a wide-array of mathematical research. Dubbed generalized hitomezashi patterns (GHPs) by Sen & Martinez, these designs are reversible and can be 'self-dual.' We investigate the possible 'dual-' or 'flip-symmetries' in GHPs; the inclusion of these symmetries is equivalent to considering two-colour symmetry groups. Following the work of Sen & Martinez, who found the wallpaper symmetry groups that are compatible with GHPs, this paper investigates the possible two-colour symmetries compatible with GHPs. We prove that there are four possible rosette two-colour symmetry types, completely describe the properties on binary strings that create these symmetries, and prove that exactly 17 of the 46 two-colour wallpaper groups are compatible with GHPs.
Source: arXiv (Cornell University)
Author: Martinez, Megan A.
Institution: Ithaca College
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204768003