Constraint Functional Logic Programming for Origami Construction

Computational Science and Its …, 2007

Computational origami is the computer assisted study of origami as a branch of science of shapes. The origami construction is a countably finite sequence of fold steps, each consisting in folding along a line. In this paper, we formalize origami construction. We model origami paper by a set of faces over which we specify relations of overlay and adjacency. A fold line is determined by a specific fold method. After folding along the fold line, the structure of origami is transformed; some faces are divided and moved, new faces are created and therefore the relations over the faces change. We give a formal method to construct the model origami. The model furthermore induces a graph of layers of faces. We give two origami examples as the application of our model. They exhibit non-trivial aspects of origami which are revealed only by formal modeling. The model is the abstraction of the implemented core of the system of computational origami called Eos (E-origami system).

2000

We present a computational origami system called Eos. Eos simu- lates what a human origamist would do with a piece of origami paper and by hands. Moreover, it assists us in reasoning about geometri- cal properties of an origami. We describe the basic capabilities of the system, which include symbolic and numeric constraints solving, automated theorem proving, visualization of an

Proceedings of the …, 2007

We describe Huzita's origami axioms from the logical and algebraic points of view. Observing that Huzita's axioms are statements about the existence of certain origami constructions, we can generate basic origami constructions from those axioms. Origami construction is performed by repeated application of Huzita's axioms. We give the logical specification of Huzita's axioms as constraints among geometric objects of origami in the language of the firstorder predicate logic. The logical specification is then translated into logical combinations of algebraic forms, i.e. polynomial equalities, disequalities and inequalities, and further into polynomial ideals (if inequalities are not involved). By constraint solving, we obtain solutions that satisfy the logical specification of the origami construction problem. The solutions include fold lines along which origami paper has to be folded. The obtained solutions both in numeric and symbolic forms make origami computationally tractable for further treatments, such as visualization and automated theorem proving of the correctness of the origami construction.

We report the progress of the development of a computational origami system with a non-trivial example of constructing a regular heptagon on our system. It simulates origami folds based on the geometrical computation of origami faces created and transformed by paper folds. With our system, origamists can construct efficiently and precisely origami on the computer and study various geometrical properties of origami as well as enjoy constructing artworks. The core of our system is a Mathematica package consisting of Mathematica functions for folding and displaying origami. The package can be loaded and executed from the Mathematica notebook in the usual way. Moreover, we developed a web interface based on webMathematica technology for accessing our package over the Internet.

Proceedings of the …, 2006

We are developing a system called Eos (E -Origami System). Eos does what a human origamist would do with a piece of origami and moreover assists in reasoning about geometrical properties of origami construction. It is a collection of Mathematica programs specializing in computational origami. It has capabilities of symbolic and numeric constraints solving, automated theorem proving, visualization of origami constructions and web interfacing. In this paper we will describe the basic features of Eos in details.

Frontiers of Computer Science in China, 2008

We present a computing environment for origami on the web. The environment consists of the computational origami engine Eos for origami construction, visualization, and geometrical reasoning, WebEos for providing web interface to the functionalities of Eos, and web service system Scorum for symbolic computing web services. WebEos is developed using Web2.0 technologies, and provides a graphical interactive web interface for origami construction and proving. In Scorum, we are preparing web services for a wide range of symbolic computing systems, and are using these services in our origami environment. We explain the functionalities of this environment, and discuss its architectural and technological features.

Electronic Proceedings in Theoretical Computer Science

This paper shows a cut along a crease on an origami sheet makes simple modeling of popular traditional basic folds such as a squash fold in computational origami. The cut operation can be applied to other classical folds and significantly simplify their modeling and subsequent implementation in the context of computational origami.

Origami paper folding has a long tradition in Japan's culture and education. The second author has recently developed a software system, based on functional logic programming and web technology, for simulating origami paper folding on the computer (the "origami computing problem" or the "forward origami problem"). This system is based on the implementation of the six fundamental origami folding steps ("origami axioms") formulated by Huzita. In this paper, we consider the problem of automatically proving general theorems on the result of origami folding sequences (the "origami proving problem") using algebraic methods, in particular the first author's Gröbner bases method. We also give some comments on finding origami folding steps that result in a desired object (the "origami solving problem" or "inverse origami problem").

2008

Computational Origami is a branch of the science of shapes, where we study computational and mathematical aspects of origami. One of the foundational studies of the computational origami is the axiomatic definition of origami foldability by Huzita in 1989. We describe Huzita's origami axioms from the logical and algebraic points of view. Observing that Huzita's axioms are statements about the existence of certain origami constructions, we can generate basic origami constructions from those axioms. Origami construction is performed by repeated application of Huzita's axioms. We give the logical specification of Huzita's axioms as constraints among geometric objects of origami in the language of the firstorder predicate logic. For this purpose, we define a many-sorted language L to give a logical formalization to Huzita's axioms and specify other geometric properties of origami elements, as well. The logical specification is then translated algebraic forms, i.e. polynomial equalities, disequalities and inequalities. If inequalities are not involved, by constraint solving we obtain solutions that satisfy the logical specification of the origami construction problem. The solutions include fold lines along which origami paper has to be folded. The obtained solutions both in numeric and symbolic forms make origami computationally tractable for further treatments, such as (1) simulation of the origami construction parallel with the graphic visualization of construction steps and (2) automated theorem proving of the correctness of the origami construction. To simulate origami construction, we model origami paper by a set of faces over which we specify relations of overlay and adjacency. After folding along the fold line, the structure of origami is transformed. Some faces are divided and moved, new faces are created and therefore the relations over the faces change. We give a formal method to construct the model origami. Using the algebraic theorem prover Gröbner basis, we prove some properties of the final origami object.

ACM Transactions on Computer-Human Interaction

Despite the recent trend of computational origami for human-computer interaction (HCI) and digital fabrication, it is still difficult for designers to complete a series of design, simulation, and fabrication of objects leveraging computational origami theory. In this paper, we propose Crane, an integrated origami design platform implemented with Grasshopper. With this platform, users can seamlessly (1) design the 2D and 3D crease pattern, (2) simulate 3D folding transformation from the given crease pattern, (3) inversely find a new pattern under design constraints, (4) thicken the 2D pattern into a 3D volume along with the appropriate hinge structures for different fabrication methods, and (5) optionally connect the resulting design to other Rhinoceros or Grasshopper plugins for post-processes. To help understand how to use our system and demonstrate its feasibility, we showed three examples of origami products designed using our system. We also reported user feedback from the works...