Cylindrical Hitomezashi Patterns

• Cylindrical hitomezashi patterns are stitching designs on a grid with a periodic (cylindrical) edge that preserves the local alternation of running stitches.
• They use binary encoding and oriented models to assign homology classes and winding numbers, ensuring each nontrivial loop winds exactly once around the cylinder.
• This formulation bridges planar and toroidal theories by incorporating overflow conditions and symmetry, enabling new combinatorial and topological classifications.

Cylindrical hitomezashi patterns are hitomezashi stitching patterns realized on a grid with one periodic direction, so that the local alternation rules of square-grid running stitch are preserved while the global topology changes from a planar rectangle to a cylinder. In the classical square-grid model, each horizontal or vertical line of stitching is determined by a single binary choice for its first visible stitch; in the most explicit recent cylindrical formulation, this data is recast in an oriented model on $\mathbb{Z}\times \mathbb{Z}/N\mathbb{Z}$, where loops acquire homology classes and nontrivial winding becomes a central invariant (Hayes et al., 2020, Seaton et al., 2022, Xie, 5 Sep 2025).

1. Foundational models and encodings

The basic hitomezashi mechanism is linewise alternation. On a square grid, each vertical line and each horizontal line is stitched as a running-stitch line, and once the first stitch on that line is chosen to be on the front or on the reverse, the rest of the line is forced by alternation. In the workshop formulation, “there are only two ways in which the first stitch in a line can be made,” so if a piece has $m$ vertical lines and $n$ horizontal lines, “the number of ways in which it could be formed is $2^{m+n}$” (Hayes et al., 2020). This produces the standard binary encoding of a hitomezashi pattern by one bit per grid line.

A more explicit mathematical specification encodes regular designs by two finite binary words $v,w\in\{0,1\}^\*$, repeated periodically in the two coordinate directions. The word $v$ specifies the vertical lines, read across the bottom, and $w$ specifies the horizontal lines, read up the left-hand side. Complementation $\overline{w}$ swaps $0\leftrightarrow 1$, reversal $\widetilde{w}$ reverses the word, and the entire front-side pattern is determined by the pair $m$0; the reverse-side pattern is encoded by $m$1 (Seaton et al., 2022). This periodic-word formalism is already adapted to quotienting by a translation, so it is naturally suited to cylindrical realizations.

For periodic surfaces, the parity-based planar definition is replaced by an oriented formulation. Let $m$2. In the oriented hitomezashi pattern $m$3, each horizontal row $m$4 receives a constant orientation determined by $m$5, and each vertical column $m$6 receives a constant orientation determined by $m$7; oriented hitomezashi paths and loops are alternating horizontal-vertical directed walks (Xie, 5 Sep 2025). This formulation removes the dependence on parity classes in the ambient coordinates and is the standard entry point for cylindrical and toroidal theories.

2. Cylindrical topology, loop types, and homology

In Xie’s convention, a cylindrical hitomezashi pattern is $m$8: the horizontal axis is infinite and the vertical axis is periodic of circumference $m$9 (Xie, 5 Sep 2025). Geometrically, this is an infinite strip with rows identified modulo $n$0, and combinatorially it is the intermediate case between planar and toroidal hitomezashi. Earlier expositions describe the same transition more informally as identifying opposite edges of a planar rectangle so that one coordinate is taken modulo an integer (Hayes et al., 2020).

The decisive new invariant on the cylinder is homology. For a loop $n$1, define

$n$2

Its homology class is $n$3. Loops of homology class $n$4 are trivial, while loops of nonzero class are nontrivial (Xie, 5 Sep 2025). The cylinder has first homology $n$5, generated by one traversal of the periodic direction, so this quantity is the winding number.

A structural observation sharply restricts the possible nontrivial classes. On a given cylindrical hitomezashi pattern, no two loops, and no loop with itself, meet transversely. Since a loop representing a nontrivial multiple of the primitive generator would necessarily have a transverse self-intersection, every nontrivial cylindrical hitomezashi loop must have homology class $n$6 (Xie, 5 Sep 2025). Thus cylindrical nontriviality is rigid: a nontrivial loop winds exactly once around the cylinder, either in the positive or in the negative direction.

The same papers also distinguish between contractible and noncontractible behavior in a more geometric language. In planar or rectangular settings, one distinguishes loops from “edge-to-edge paths”; once a boundary pair is identified to form a cylinder, some of those paths become closed loops or winding curves (Hayes et al., 2020). This suggests a useful practical dichotomy: local stitching rules are unchanged by the quotient, but global loop topology is altered by the periodic identification.

3. The main cylindrical classification theorem

The central combinatorial theorem for cylindrical hitomezashi patterns is expressed in terms of partial sums and “overflowing” subsequences. For a cyclic vertical encoding $n$7, one lifts it periodically to $n$8 and defines interval sums $n$9. When $2^{m+n}$0, the extreme partial sums are finite and satisfy

$2^{m+n}$1

This range $2^{m+n}$2 measures the maximal vertical imbalance that can be absorbed without forcing a loop to use the cylindrical topology (Xie, 5 Sep 2025).

For a cylindrical loop $2^{m+n}$3, one defines its bounding cylinder as the smallest strip

$2^{m+n}$4

containing $2^{m+n}$5, and its bounding sequence as the finite horizontal subsequence $2^{m+n}$6 (Xie, 5 Sep 2025). A finite subsequence $2^{m+n}$7 is positively overflowing if $2^{m+n}$8, minimally positively overflowing if this holds and no proper contiguous subsequence is positively overflowing, with the negative version defined analogously.

Xie’s main theorem states that, assuming $2^{m+n}$9, cylindrical loops of homology $v,w\in\{0,1\}^\*$0 are in bijection with minimally positively overflowing subsequences, and cylindrical loops of homology $v,w\in\{0,1\}^\*$1 are in bijection with minimally negatively overflowing subsequences. More precisely: for any cylindrical hitomezashi loop $v,w\in\{0,1\}^\*$2 of homology $v,w\in\{0,1\}^\*$3, its bounding sequence is minimally positively overflowing; conversely, every minimally positively overflowing subsequence $v,w\in\{0,1\}^\*$4 determines a unique cylindrical hitomezashi loop of homology $v,w\in\{0,1\}^\*$5, with the corresponding negative statement for homology $v,w\in\{0,1\}^\*$6 (Xie, 5 Sep 2025).

This theorem is the cylindrical analogue of Pete’s planar loop description. In the plane, a loop is controlled by two finite balanced encodings of equal height; on the cylinder, a nontrivial loop is detected by a one-sided overflow condition in one encoding relative to the range of the other (Xie, 5 Sep 2025). The shift from matched excursions to overflow is the characteristic combinatorial signature of cylindrical hitomezashi.

There is also a degenerate regime. If $v,w\in\{0,1\}^\*$7, then there are no nontrivial cylindrical hitomezashi loops, and there exists an infinite cylindrical hitomezashi path that contains a horizontal edge $v,w\in\{0,1\}^\*$8 for each $v,w\in\{0,1\}^\*$9 (Xie, 5 Sep 2025). In that case the net vertical bias prevents a once-winding loop from closing.

4. Planar antecedents and toroidal consequences

Cylindrical theory is built on two planar results. First, Pete’s theorem identifies planar hitomezashi loops with pairs of Dyck paths of the same height; equivalently, with pairs of opposite excursions of the same height in the encoding sequences (Defant et al., 2022, Xie, 5 Sep 2025). Second, Defant and Kravitz proved that every planar hitomezashi loop has odd width and odd height, length congruent to $v$0, and area congruent to $v$1 (Defant et al., 2022). Ren and Zhang later gave a shorter proof of the mod-$v$2 length theorem by introducing hitomezashi excursions and proving that every loop length is congruent to $v$3 (Ren et al., 2023).

The planar proofs do not transfer automatically to the cylinder. Ren and Zhang’s argument relies on a leftmost vertical line, a half-plane decomposition, and a noncrossing lemma in a simply connected region, so the proof uses ingredients that are absent on a cylinder with one periodic direction (Ren et al., 2023). Defant and Kravitz’s area argument also depends on a bounded interior region and Pick’s theorem (Defant et al., 2022). A plausible implication is that contractible cylindrical loops should inherit the planar congruences by lifting to the universal cover, but this is presented as conjectural or heuristic rather than as a theorem in those papers (Defant et al., 2022, Ren et al., 2023).

Toroidal work supplied the precursor to the modern cylindrical formulation. Ren and Zhang defined toroidal hitomezashi patterns via the same orientation data used later on cylinders, proved modular constraints on loop lengths, and obtained optimal residual information on loop counts and homology classes (Ren et al., 2023). Xie then used cylindrical lifting and the overflowing-subsequence theorem to complete Ren–Zhang’s classification in the balanced toroidal case. If $v$4, then comparison of the ranges $v$5 and $v$6 decides the possible nontrivial homology classes: if $v$7, vertical classes $v$8 occur; if $v$9, no nontrivial loops occur; if $w$0, horizontal classes $w$1 occur (Xie, 5 Sep 2025).

The toroidal counting theorem is likewise completed by the cylindrical analysis. In the fully unbalanced case $w$2 and $w$3, the number of nontrivial loops is $w$4, and they all share the same primitive homology class up to sign (Xie, 5 Sep 2025, Ren et al., 2023). In the mixed and balanced cases, the number of nontrivial loops is expressed through counts $w$5 of minimal overflowing arcs. Cylindrical hitomezashi is therefore not only an intermediate geometry but also the technical mechanism that closes the toroidal classification.

5. Symmetry, duality, friezes, and periodic design families

Duality is intrinsic to hitomezashi because the reverse side is forced. In the binary-word formalism, the back-side pattern is encoded by $w$6, and a pattern is self-dual up to translation when the complemented words are cyclic shifts of the originals (Seaton et al., 2022). This periodicity language is especially natural on a cylinder, where a shift becomes a rotation around the periodic direction. The Pell persimmon polyomino patterns provide the most explicit family: defining Pell words $w$7 by a recursion involving complement and reversal, then taking

$w$8

produces periodic, self-dual patterns whose fundamental period is $w$9, where $\overline{w}$0 is the Pell number. The associated “Persimmon-Snowflake Conjecture” states that the largest polyomino in the Pell persimmon pattern of order $\overline{w}$1 is the Fibonacci snowflake of order $\overline{w}$2 (Seaton et al., 2022). Because these designs are specified by periodic words, they admit immediate cylindrical quotients by taking the circumference to match the word period.

A complementary symmetry framework comes from two-sided friezes. Hitomezashi strips can be treated as two-sided friezes whose front and back are complementary patterns, encoded by a periodic vertical word $\overline{w}$3 and a finite horizontal word $\overline{w}$4, with the back determined by $\overline{w}$5 (Seaton, 30 Jan 2026). Cromwell’s two-sided frieze taxonomy has 31 groups; for hitomezashi, 18 are obstructed by complementarity arguments and 13 are realized by explicit examples (Seaton, 30 Jan 2026). Under cylindrical wrapping, translation along the frieze axis becomes rotation around the cylinder, and screw symmetries become helical symmetries. This supplies a direct route from frieze-group classification to cylindrical symmetry types.

The square grid is not the only available geometry. The dilute hitomezashi model on the isometric grid keeps only every second line in each of the three lattice directions and imposes that each visited vertex has degree two, so the stitched design is a disjoint union of closed loops (Seaton, 28 Feb 2025). Periodic words such as $\overline{w}$6, $\overline{w}$7, $\overline{w}$8, and the recursively defined Koch words yield wallpaper patterns with symmetry groups $\overline{w}$9 and $0\leftrightarrow 1$0, and the paper explains that such wallpaper patterns can be wrapped onto a cylinder by choosing a circumference equal to one period or an integer multiple (Seaton, 28 Feb 2025). This is not square-grid hitomezashi, but it demonstrates that cylindrical adaptation is a general periodic-quotient operation rather than an artifact of one lattice.

A common misconception is that cylindrical hitomezashi is merely a decorative reformulation of planar patterns. The literature shows otherwise. Once one coordinate is periodic, loop classes, nontrivial winding, bounding cylinders, overflowing subsequences, screw symmetries, and front/back helical interpretations all become mathematically substantive features (Xie, 5 Sep 2025, Seaton, 30 Jan 2026, Seaton, 28 Feb 2025).

6. Scope, limitations, and open directions

The literature distinguishes clearly between what is proved for cylinders and what is only suggested by planar analogy. The 2020 introduction to sashiko treats hitomezashi primarily as a two-dimensional pattern on a square grid and states that many ingredients adapt naturally to a cylindrical setting, but it does not formalize cylindrical topology (Hayes et al., 2020). The 2023 toroidal paper explicitly listed the infinite cylindrical grid $0\leftrightarrow 1$1 as an open problem (Ren et al., 2023). Xie’s 2025 paper resolved a large part of that program for nontrivial cylindrical loops, but it did so in the oriented model and for the specific homological phenomena governed by overflowing subsequences (Xie, 5 Sep 2025).

Several open directions remain explicit in the record. The two-sided frieze paper notes that a full two-sided wallpaper treatment would invoke the eighty layer groups and places that beyond scope (Seaton, 30 Jan 2026). Ren and Zhang’s short proof isolates the topological obstacles to extending planar mod-$0\leftrightarrow 1$2 arguments to the cylinder, especially for winding loops (Ren et al., 2023). The binary-word paper leaves the Persimmon-Snowflake Conjecture unresolved (Seaton et al., 2022). The isometric-grid paper describes the dilute setting as promising precisely because the fully packed version is hard to understand (Seaton, 28 Feb 2025).

The present state of the subject therefore has a definite shape. Cylindrical hitomezashi patterns are now understood as periodic quotients of line-encoded or oriented hitomezashi systems; nontrivial loops are classified by homology $0\leftrightarrow 1$3 and by minimal overflow conditions; toroidal homology classification and counting reduce in essential part to cylindrical analysis; and symmetry is organized by periodic words, complement duality, frieze groups, and lattice-periodic wrapping (Xie, 5 Sep 2025). What remains less settled is the extension of planar loop congruences to noncontractible cylindrical loops, the global classification of cylindrical self-dual families beyond the currently explicit word constructions, and the systematic treatment of cylindrical hitomezashi outside the square-grid setting.