A Whole New Side: The Duality Symmetries of Hitomezashi

[2608.27542] A Whole New Side: The Duality Symmetries of Hitomezashi Skip to main content Search Submit Donate Log in Mathematics > History and Overview arXiv:2608.27542 (math) [Submitted on 27 Aug 2026] Title: A Whole New Side: The Duality Symmetries of Hitomezashi Authors: Megan A. Martinez View PDF HTML (experimental) 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. Subjects: History and Overview (math.HO) MSC classes: 00A66 Cite as: arXiv:2608.27542 [math.HO] (or arXiv:2608.27542v1 [math.HO] for this version) https://doi.org/10.48550/arXiv.2608.27542 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Megan Martinez [ view email ] [v1] Thu, 27 Aug 2026 16:21:04 UTC (21,548 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.HO < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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