Abstract Traditional origami design is generally based on designers’ artistic intuition and skills, mathematical calculations, and experimentations, which can involve challenges for crease patterns with a large number of vertices. To develop novel origami structures for engineering applications, systematic and easy-to-implement approaches capable of generating diverse origami patterns are desired, without requiring extensive artistic skills and experience in origami mathematics. Here, we present a computational method for automatically assigning mountain-valley fold lines to given geometric configurations of origami structures. This method is based upon a geometric-graph-theoretic representation approach combined with a graph-theoretic cycle detection algorithm, taking the subgraphs of a given structure as inputs. Then, a mixed-integer linear programming (MILP) model is established to find flat-foldable origami patterns under given constraints on the local flat-foldability and degree of vertices, leading to the identification of crease lines associated with local minimum angles. Numerical examples are presented to demonstrate the performance of the proposed approach for a range of origami structures with degree-4 or -6 vertices represented by their corresponding subgraphs.

中文翻译：

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基于图论和混合整数线性规划的可折叠折纸图案的山谷折线分配

摘要 传统折纸设计一般基于设计师的艺术直觉和技巧、数学计算和实验，这可能涉及具有大量顶点的折痕图案的挑战。为了开发用于工程应用的新颖折纸结构，需要能够生成多种折纸图案的系统且易于实施的方法，而无需在折纸数学方面具有广泛的艺术技能和经验。在这里，我们提出了一种自动将山谷折叠线分配给折纸结构的给定几何配置的计算方法。该方法基于几何图论表示方法与图论循环检测算法相结合，将给定结构的子图作为输入。然后，建立了混合整数线性规划 (MILP) 模型，以在给定局部平面可折叠性和顶点度数的约束下找到可平面折叠的折纸图案，从而识别与局部最小角度相关的折线。数值例子展示了所提出的方法对于一系列折纸结构的性能，这些折纸结构的度数为 4 或 -6 顶点由其相应的子图表示。