Hitomezashi is a traditional form of Japanese embroidery in which patterns arise from the alignment of single stitches made on a grid, as in the picture above. A random Hitomezashi pattern on a rectangle can be generated by choosing two random sequences in a two-symbol alphabet, for example 10111100011 and 101001110. The first sequence governs the vertical stitches and is written along the horizontal axis from left to right, and the second sequence governs the horizontal stitches and is written along the vertical axis from bottom to top. Every vertical and horizontal line in the grid consists of an alternating pattern of stitches. The symbol (i.e., 0 or 1) corresponding to each row and column of stitches determines whether the stitches in that row or column appear in the odd positions or the even positions.
The part about homology classes of loops on the torus is algebraic topology, which is part of topology, although the original problem about the grid is combinatorics.
Thanks a lot! Do you know any accessible/ easy books treating these topics?
No, I’m not aware of any books on this, but the YouTube video I linked to in the post is worth watching, if you haven’t seen it already.
Good morning, where can I find more articles regarding this tonic objects and related applications? Is this Topology?
The part about homology classes of loops on the torus is algebraic topology, which is part of topology, although the original problem about the grid is combinatorics.
The paper by Defant and Kravitz is also online at https://arxiv.org/abs/2201.03461
Some other relevant papers that I didn’t mention in the article are mentioned in the bibliography of the paper by Ren and Zhang:
Gábor Pete, “Corner percolation on Z2 and the square root of 17.” The Annals of Probability, 36(5), 2008.
Colin Defant, Noah Kravitz, Bridget Eileen Tenner, “Extensions of Hitomezashi Patterns”. https://arxiv.org/abs/2208.14428
Wikipedia has a page on Hitomezashi and related embroidery techniques: https://en.wikipedia.org/wiki/Sashiko