Rethinking Origami: A Generative Specification of Origami Patterns with Shape Grammars

Highlights

• Rational method for generating origami patterns.

• Modeling technique for designing origami tessellations.

• Shape Machine, new shape grammar interpreter that automates shapes calculation to create new patterns.

• Origami pattern considered as a combination of different shapes.

Abstract

As a ubiquitous paper folding art, origami has promising applications in science and engineering. Many software and parameterized methods have been proposed to draw, analyze and design origami patterns. Here we focus on the shape grammar formalism and the Shape Machine, a shape grammar interpreter that has managed to automate the seamless shape calculations that the shape grammar formalism advocates. Different from other origami pattern generation methods, shape grammars generate origami patterns through recursive applications of shape rewriting rules using lines and curves. Based on this concept, the transformations between some common origami patterns are reorganized following visual cues and reasoning. Four examples of generating origami pattern are presented to show the capability of Shape Machine in origami design, including construction, modification and programming of an origami pattern. The new origami designs inspired by this work prove that shape grammars and Shape Machine provide a perspective and modeling technique for creating origami tessellation patterns.

Introduction

Non-representational origami, e.g. origami tessellation, has a deep history in various fields including art [1], design [2], architecture [3], computational geometry [4] and mathematics [5]. Classic origami tessellations, such as the Miura-ori [6], the Resch pattern [7] and so forth, routinely appear in several pattern books showcasing focused studies in the architecture of form. A sample of such origami tessellations is shown in Fig. 1 all generated within the Shape Machine, the formal modeling system discussed in this paper.

The scientific study of the origami tessellation can be traced back to the study of the origami design, itself kicked off after the pioneering work of the computer scientist and artist Ron Resch [7], [8] who began designing and folding paper forms using mathematical and computational algorithms. Since then, scientists and engineers have been following his paradigm and have established origami research as a growing body of research with its own structures, forms and mathematical laws governing origami design [4], [5], [9], [10] as well as a growing number of novel applications in various fields and primarily in science and engineering. Examples of such applications of the geometry of origami includes the design of metamaterials [11], [12], the DNA folding in nanotechnology [13], [14], origami acoustic metamaterials [15], graphene folding in nanotechnology [16], deformable optoelectronic and thermoset shape-memory polymer applications [17], [18], tunable structures [15], [19], and deployable structures [20], [21]. As the applications of origami design get broader exposure, advances in origami mathematics and origami computation technologies bring the contemporary design of such patterns to a new level of complexity.

Current software developed for origami design and analysis include the Treemaker [22], Freeform Origami [23], and Merlin [24], among others. The software TreeMaker is based on the powerful design techniques of circle-river packing. It can construct the full crease pattern for a wide variety of origami bases. The patterns constructed in TreeMaker are commonly the most efficient solutions possible for a given tree figure. Freeform Origami enables a freeform variation of rigid-foldable structure by introducing bidirectionally flat-foldable planar quadrilateral (BDFFPQ) meshes. It can also generate the initial pattern suited for almost any target 3D form [10]. Merlin is a software for nonlinear structural analysis of both rigid and non-rigid origami assemblages. Based on the nonlinear mechanics, it uses the bar-and-hinge model for origami discretization.

Powerful as they may be, these formal representations of origami models are primarily analytical or numerical, and they do not lend themselves easily in the creation of origami designs or the comparative generation and evaluation of classes of origami designs. Take for example, the criteria for folding of a flat pattern without self-intersections: Such rigidly flat foldable tessellations can be evaluated using several theorems including the Maekawa–Justin Theorem, the Kawasaki–Justin theorem, the Even Degree Theorem, the Local Min Theorem and the fold-angle multiplier, all used to evaluate the foldability of a single vertex [4]. For a simple degree-4 vertex, all possible assignments of mountain and valley creases can be enumerated and tested if the given assignment is valid. However, the situation becomes much more complicated when the networks of creases consist of multiple vertices. Because different vertices may place contradictory conditions upon other creases, several collisions between layers of paper may happen. In this case, additional mathematical arguments such as the Justin Isometry Theorem, the Justin Non-twist Theorem, vector formulations and the local Flat-foldability Graph [5], may need to be deployed to decide whether locally flat foldability can be obtained. And still, there is no guarantee that the entire creased pattern folds flat without self-intersection. As the origami folding process involves non-rigid deformation and curve creases, the mathematical laws become much more complicated, and most of the times, no final analytical results can be found.

A very different approach emphasizing the constructive specification of origami designs employs symbolic grammars or rewrite production systems, such as the L-system (Lindenmayer system) [25], a production system that captures the generation of plant cells and self-similar fractals and periodic topology [26]. Specific classes of origami patterns, for example, the Heighway dragon curve origami pattern, can be generated by a set of parallel rewrite rules in the manner of an L-system simulating a fractal-like folding of a strip of paper [27]. Still, useful as this specification may be, it requires facility with computer programming and remains inaccessible to those who are not familiar with Python or other programming languages.

A different approach focuses on the comparative study of the geometry of classes of origami designs and tessellations [28]. This approach gives an organization of several rigidly foldable origami tessellations and examines how they are related. Because of the periodicity of origami tessellation, only one or two basic units of an origami pattern need to be illustrated and the rest can be derived from them: ‘Vertex mirror symmetry’, ‘Vertex inversion’ or ‘collapse to degree six vertices’ are frequently mentioned to transform one pattern to another [28]. Still, to designers without an engineering background, it is difficult to clearly understand the transformation through geometric modification. A typical transformation from an Arc pattern to a Miura-ori pattern is shown as an example, in Fig. 2. Both patterns are well defined in origami mathematics to ensure the rigidly flat foldability of the plane [9]. Based on the given geometry, the injunction ‘Add vertices inversion’ means that the points A, B and C in the Arc pattern should be moved to the points ${\mathrm{A}}^{\prime }$, ${\mathrm{B}}^{\prime }$ and ${\mathrm{C}}^{\prime }$, which are symmetric about an axis ${\mathrm{MM}}^{\prime }$. This also means that the corresponding mountain and valley assignments have to be changed. In all, the representation, interpretation and evaluation of such new origami designs through this geometric method still require a deep sophistication in geometric modeling techniques and the validity of the results is not guaranteed. We need an alternative perspective to rethink origami pattern.

The methods mentioned above generate origami patterns, mathematically and parametrically. These methods are powerful and efficient, but require complex modeling methods and a steep learning curve to understand the appropriate tools and methodologies. More detrimentally, any changes and modifications to the original design require a complete restructuring of the whole pattern. As the origami system becomes non-flat foldable and non-rigid, the design process becomes cumbersome, difficult and uncertain. Still, since rational thought cannot predict everything in advance, seeing and drawing may work perfectly for origami pattern generation. In other words, if we consider origami patterns as shapes (arrangements of lines and arcs), the description, interpretation and evaluation of origami design can all be done by visual reasoning. Here, shape grammars [29], [30], [31], [32], [33] are adopted to design origami patterns.

Shape grammars are production systems using shape rewriting rules to perform computations with shapes. Their uniqueness with respect to all formal approaches in computational design is that they operate exclusively with shapes rather with some other symbols i.e. numbers, text, or symbolic instructions in some programming language. Their formidable formalism relying on the algebras of shapes $U_{ij}$, the algebras of labeled shapes $V_{ij}$, and the algebras of weights $W_{ij}$, for $i\le j$ and $j\le 3$, and the intuitive construct of a visual rule in the form of a pair of shapes, labeled shapes or weights defined in any of the algebras above, have provided a strong foundation for formal studies in design, and an unwavering commitment to visual reasoning. Their very reliance on visual aspects of form – and the ways these enter in visual calculations – has made them one of most accessible and intuitive formalisms to use in formal studies in design. Several applications have been developed in various fields over the years including architectural design [34], landscape architecture [35], engineering [36], painting [37], furniture design [38], ornamental design [39], origami design [40], and others. Still, this resolute commitment of shape grammars in shapes and shape rules does not come without its toll: shapes consisting of lines, planes and solids fuse every time they are combined in a calculation to produce new and unexpected results and despite a long and sustained effort to produce a shape grammar interpreter to perform in a computer system the shape calculations that are possible with a pencil and a paper, all efforts have been inconclusive.

A sophisticated, but ultimately severely constrained, approach using a combinatorial calculation of the boundaries of the shapes to predict emergent shapes, has provided some useful applications, albeit all failing to provide a general solution to the problem [41]. Still, the situation is not as grave as it may seem. The single major obstacle to take on is the implementation of shape embedding [42], that is, the implementation of the mathematical concept of the “part relation” between two shapes, and it appears that the riddle has finally been solved: the very first software technology that has successfully managed to put the formalism in practice is the Shape Machine [43], a new computational technology that features a new way to specify the way geometric shapes are digitally represented, indexed, queried and operated upon. In this work, existing and new origami patterns are modeled, modified and programmed in Shape Machine and along the way some preliminary thoughts are discussed on the future of this technology in origami design.