Origami is a form of art, culture, and play in which a diversity of shapes are created by folding a sheet of paper. In recent years it has outgrown the boundaries of Japanese culture to become a universal artistic medium. While its form is based on strict rules under which a single piece of paper must be used and no cutting or stretching is allowed, origami has tested a diverse range of expressions, from abstract to concrete, simple to complex, two-dimensional to three-dimensional, and static to dynamic. Origami is also gaining attention as a new method in engineering design for its dynamic characteristics - its capacity to allow for change between two- and three-dimensional forms - the property of the continuity of one sheet of paper, and the quality of the shadows and curves created by folding.
In "Architectural Origami," we propose an origami design theory for generating adaptive environments by fully utilizing the inherent functionality, dynamic characteristics, and diversity of origami as a method of problem solution in an engineering context. The three characteristics of this theory are (1) solving inverse problems in which origami shapes are generated based on various conditions, rather than origami being regarded as a shape and applied to architectural problems; (2) creating frameworks that enable the process of discovery of new meanings and relationships through interaction between designer and shape, rather than just explicitly obtaining one shape from conditions when solving inverse problems; and (3) creating physical spaces that until now could not be realized by giving substance to computationally generated abstract origami. We will present these through three new kinds of origami theory and application: "Origamizing 3D Surfaces," "Freeform Origami," and "Rigid Origami."
In "Origamizing 3D Surfaces," a design theory that "origamizes" any kind of three-dimensional form; that is to say, a theory that enables the realization of arbitrary polyhedra simply by folding a square shape is constructed. We have constructed the system "origamizer," that calculates crease patterns for folding various given three-dimensional shapes by computationally solving the inverse problem of the technique to tuck-fold paper to form a three-dimensional shape. Moreover, using this method has enabled for the first time the realization of polygon mesh shapes such as the "Utah Teapot" and "Stanford Bunny" - standard computer graphics objects - using origami.
In "Freeform Origami," by using an interactive design system that explores continuously and reversibly within the solution space of forms attainable by folding paper - that is to say, possible forms of origami - the geometric world of origami, which contains different physical metaphors from the real physical world, is revealed. Because in the physical world the procedure of folding paper is a discontinuous and irreversible process that involves the destruction of materials, searching freely for possible forms is difficult. Thus we perform simulations of a virtual material, "Freeform Origami," whose behavior is based on the geometric constraints that a shape created by folding paper should retain and not the physical constraint conditions of paper. "Freeform Origami" responds to a force applied to it and changes its shape to a special three-dimensional surface that can always be folded from one developed surface and be folded to a flat state. Since all of the crease patterns generated in real time in this system are valid solutions, the corresponding three-dimensional shape can be reproduced by plotting the pattern on paper and then folding the paper along it.
In "Rigid Origami," the geometric concepts concerning the deformation mechanisms of origami are extracted and translated again to a physical space, thereby constructing a new transformable structure with a different size (scale) and materials. The mechanism that flexibly transforms origami comprises material flexibility and geometric flexibility. Transformation models that describe mechanisms using only the latter - in other words, mathematical models in which rigid facets are connected to each other by the axis of rotation on the fold lines - are called "Rigid Folding." Rigid-foldable origami is in fact extremely limited, and it is known that most of known origami patterns utilize stretching and shrinking of materials. In contrast to this, if rigid folding models can be found and translated to the physical world, it is possible to create transformable structures heretofore undiscovered. Thus by constructing the geometric space of rigid-foldable origami and searching for/discovering special shapes that, while complex, generate non-trivial one-DOF folding motion, then adding thickness that is consistent with the structure, we were able to generate a new human-scale transformable structure made of thick panels.
Origami is a formative activity that creates various shapes by transforming a sheet of paper according to the rule "no stretching or shrinking; no cutting or pasting." Compared with other formative media such as sculpture, molding, and paper craft, the rules imposed on origami are strict, and the geometric constraints generated by these strict rules give origami shapes their distinctive characteristics. Such geometric constraints tend to be viewed as elements that interfere with free form, but in truth they have the opposite effect, serving as the source for rich expressive potential and generating portrayal and suggestive diversity. For example, with origami it is not possible to use methods for assembling detail that are like sculpturally repeating additions to information; in order to create a detailed form, it is necessary to modify the whole structure. Origami works that have been designed by resolving this type of composite problems have a harmonious pattern in which the whole and parts interlock.
On second thought, geometric constraints can be said to be the source of form for any and all formative media to begin with, regardless of differences in the level of their severity. This is because if a situation where there were no constraints and all possibilities were guaranteed were to be imagined, that would become random noise without shape to provide and hint of awareness or form. Geometric constraints are the structure of relationships and do not themselves have shape, but are expressed in specific shapes through external actions or conditions. Consequently, form is an action that expresses shape through externals actions such as the contrivances of the creator from the geometric constraints underlying the medium or system. For example, the buckling phenomenon* when a crumbling, crushing distortion is applied to a system comprising paper that physically contains the geometric relationships of three-dimensional curves that do not stretch or shrink generates the "ori (fold)" form, which has the differential discontinuity that is the basis of the origami shape.
The level of symmetry and irregularity of forms differ according to the strength of the geometric constraints against the degree of freedom of definable spaces. If the constraints are too strong compared to the defined degree of freedom, symmetry becomes dominant, falling into a simple form with meager variation; if they are too weak, the form will diverge into random noise whose shape is unreadable. The balance between degree of freedom and constraint is an important element for form. Consequently, under the current conditions of computational design, which has overwhelmingly increased the degree of freedom of definable spaces, in order to construct spaces with unprecedented new meaning, a need to pay high attention to geometric constraints and external conditions has been born.
From the perspective of spatial creation, an information science field that has developed in recent years, computational origami, holds new potential because it is believed that the strength of the geometric constraints inherent in origami can be balanced with increases in the degree of freedom through computation, constructing a diverse solution space. The results of research on these spatial shape aspects of computational origami are presented in this "Architectural Origami" exhibition.
In architectural origami, the solution spaces of forms generated from defined degree of freedom and underlying geometric constraints are constructed within information environments in intuitive forms, revealing origami's possible world. This possible world contains as rules geometric structures that have a duality with the rules of the physical world. If a form exists in the possible world, there is the commutativity that the form could also be constructed in the physical world. Moreover, properties that can only be confirmed through the dynamic manipulation of actually folding a piece of paper in the physical and simulational worlds are in the possible world naturally fulfilled as properties inherent in the abstract material of freeform origami. We wish to propose the internal geometric system guaranteed by such commutativity with the physical world as a methodology for exploring possible spaces.
*Buckling phenomenon: Fracture behavior in which a structure to which force has been applied loses its shape while distorting in a different direction to that of the force applied. In one dimension, it is the buckling phenomenon that occurs when a long, thin stick is compressed in an axial direction.
