COMBINATORICS AND ALGEBRA OF 1 × n CYCLIC STAMP FOLDING

This paper investigates the algebraic properties of cyclic stamp folding, contrasting them with those of traditional stamp folding. While the set of partially folded states for standard stamp folding forms a Heyting algebra, the set of partially folding states of the cyclic stamp folding under a partial order based on contacting faces forms only a join-semilattice, as a “meet” operation is not universally defined. This incomplete structure necessitates the search for a more descriptive algebraic framework. To achieve this, we analyzed the combinatorial structure of the folded states and identified a minimal generating set composed of two fundamental patterns: “stars” (alternating mountain-valley folds) and “trees” (consecutive mountain or valley folds). Any valid folded state can be constructed from a combination of these basic components. Based on this generating set, we define a category for the set of all final folded states. We demonstrate that this category is a monoidal category, where the tensor product allows for the construction of complex folded states from simpler ones. This provides a novel and more comprehensive algebraic structure for cyclic stamp folding.