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Maximal origami flip graphs of flat-foldable vertices: properties and algorithms

Thomas C. Hull, Manuel Morales, Sarah Nash, Natalya Ter-Saakov

TL;DR

This work analyzes the combinatorics of valid mountain–valley assignments for flat-foldable single-vertex crease patterns via face flips, focusing on the equal-angle vertex ${A_{2n}}$. It establishes ${OFG}(A_{2n})$ as the maximal origami flip graph among degree-${2n}$ flat vertex folds, proves its connectivity with two ${O(n^2)}$ traversal algorithms, and shows its diameter is ${n}$. It also derives a closed-form formula for the number of edges and a full degree sequence, revealing rich underlying structure and prompting several open questions about embeddings and partition-based combinatorics. These results deepen understanding of the flip dynamics and may guide design and analysis of origami-inspired metamaterials.

Abstract

Flat origami studies straight line, planar graphs $C=(V,E)$ drawn on a region $R\subset\mathbb{R}^2$ that can act as crease patterns to map, or fold, $R$ into $\mathbb{R}^2$ in a way that is continuous and a piecewise isometry exactly on the faces of $C$. Associated with such crease pattern graphs are valid mountain-valley (MV) assignments $μ:E\to\{-1,1\}$, indicating which creases can be mountains (convex) or valleys (concave) to allow $R$ to physically fold flat without self-intersecting. In this paper, we initiate the first study of how valid MV assignments of single-vertex crease patterns are related to one another via face-flips, a concept that emerged from applications of origami in engineering and physics, where flipping a face $F$ means switching the MV parity of all creases of $C$ that border $F$. Specifically, we study the origami flip graph ${\rm{OFG}}(C)$, whose vertices are all valid MV assignments of $C$ and edges connect assignments that differ by only one face flip. We prove that, for the single-vertex crease pattern $A_{2n}$ whose $2n$ sector angles around the vertex are all equal, ${\rm{OFG}}(A_{2n})$ contains as subgraphs all other origami flip graphs of degree-$2n$ flat origami vertex crease patterns. We also prove that ${\rm{OFG}}(A_{2n})$ is connected and has diameter $n$ by providing two $O(n^2)$ algorithms to traverse between vertices in the graph, and we enumerate the vertices, edges, and degree sequence of ${\rm{OFG}}(A_{2n})$. We conclude with open questions on the surprising complexity found in origami flip graphs of this type.

Maximal origami flip graphs of flat-foldable vertices: properties and algorithms

TL;DR

This work analyzes the combinatorics of valid mountain–valley assignments for flat-foldable single-vertex crease patterns via face flips, focusing on the equal-angle vertex . It establishes as the maximal origami flip graph among degree- flat vertex folds, proves its connectivity with two traversal algorithms, and shows its diameter is . It also derives a closed-form formula for the number of edges and a full degree sequence, revealing rich underlying structure and prompting several open questions about embeddings and partition-based combinatorics. These results deepen understanding of the flip dynamics and may guide design and analysis of origami-inspired metamaterials.

Abstract

Flat origami studies straight line, planar graphs drawn on a region that can act as crease patterns to map, or fold, into in a way that is continuous and a piecewise isometry exactly on the faces of . Associated with such crease pattern graphs are valid mountain-valley (MV) assignments , indicating which creases can be mountains (convex) or valleys (concave) to allow to physically fold flat without self-intersecting. In this paper, we initiate the first study of how valid MV assignments of single-vertex crease patterns are related to one another via face-flips, a concept that emerged from applications of origami in engineering and physics, where flipping a face means switching the MV parity of all creases of that border . Specifically, we study the origami flip graph , whose vertices are all valid MV assignments of and edges connect assignments that differ by only one face flip. We prove that, for the single-vertex crease pattern whose sector angles around the vertex are all equal, contains as subgraphs all other origami flip graphs of degree- flat origami vertex crease patterns. We also prove that is connected and has diameter by providing two algorithms to traverse between vertices in the graph, and we enumerate the vertices, edges, and degree sequence of . We conclude with open questions on the surprising complexity found in origami flip graphs of this type.
Paper Structure (6 sections, 1 theorem, 18 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 6 sections, 1 theorem, 18 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and maximality of ${\rm{OFG}}(A_{2n})$
  3. Connectivity of ${\rm{OFG}}(A_{2n})$
  4. Diameter of ${\rm{OFG}}(A_{2n})$
  5. Counting edges of ${\rm{OFG}}(A_{2n})$
  6. Conclusion

Key Result

Corollary 1

The flip graph ${\rm{OFG}}(A_{2n})$ is connected.

Figures (3)

  • Figure 1: The crease patterns (a) $A_4$ and (b) $A_6$ along with their origami flip graphs (for ${\rm{OFG}}(A_6)$ only the vertices with $M-V=-2$ are shown). Each vertex is labeled with the valid MV assignment to which it corresponds (bold/non-bold means mountain/valley, respectively).
  • Figure 2: Other flat vertex folds $B_4$ and $C_4$ of degree 4 and their origami flip graphs, viewed as subgraphs of ${\rm{OFG}}(A_4)$.
  • Figure 3: An example of a shwoop sequence of face flips that converts $\mu$ to $\nu$.

Theorems & Definitions (10)

  • proof
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  • Corollary 1
  • proof
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  • proof