Tetsuo Ida

University of Tsukuba · Department of Computer Science

We discuss an abstraction of origami.We formalize origami by the abstract rewriting. We describe an origami construction by abstract rewritings of a rewriting system (O, ↬), where O is the set of abstract origamis and ↬ is a binary relation on O, that models a fold. An abstract origami is a structure (∏, ∼ , ≻), where ∏ is a set of faces constituti...

We discuss an origami geometry based on Huzita-Justin’s basic folds. In the 2D Euclidean geometry, we use a straightedge and a compass to construct geometric objects. We show that Huzita-Justin’s basic folds can construct them without such tools but by hand. We reformulate Huzita-Justin’s fold rules by giving them precise conditions for their use....

We discuss the basic fold rules of origami. With a small number of fold rules, we can construct geometric shapes that we see at geometry classes of schools. We construct those shapes usually by a straightedge and a compass, so-called a Euclidian tool of construction. We explain the set of the basic fold rules and show, by examples, that it is as po...

Origami is the art of paper folding, providing the methodology of constructing a two-dimensional (2D) or three-dimensional (3D) object out of a sheet of paper solely by folding by hand.

In this chapter, we see Huzita-Justin folds HO from a logical point of view. We give the logical specification of HO as constraints among geometric objects of origami in the language of the first-order predicate logic. The logical specification is next translated to logical combinations of algebraic expressions, i.e., polynomial equalities, unequal...

We discuss computer-assisted construction and automated verification of regular polygonal knots by origami. Given a rectangular origami or a tape of an adequate length, we can construct the simplest knot by three folds. We can make the shape of the knot a regular pentagon if we fasten the knot rigidly. We analyze the knot fold formally so that we c...

In this book, origami is treated as a set of basic geometrical objects that are represented and manipulated symbolically and graphically by computers. Focusing on how classical and modern geometrical problems are solved by means of origami, the book explains the methods not only with mathematical rigor but also by appealing to our scientific intuit...

We discuss the verification in the origami geometry. In the preceding chapters, we discussed the origami construction using origami language ORIKOTO. During the construction, the logical formulas that describe the geometric configuration are formed and stored. We use those formulas for verifying the geometric properties of the constructed origami....

We present a generalization of mathematical origami to higher dimensions. We briefly explain Huzita-Justin's axiomatic treatment of mathematical origami. Then, for concrete-ness, we apply it to origami on 3-dimensional Euclidean space in which the fold operation consists of selecting a half-plane and reflecting one half-plane across it. We finally...

This short note describes the first step of the application of the geometric algebra (GA) to the computational origami system called Eos. Main results are the formalization of GA in Isabelle/HOL and the re-statement of Huzita's basic fold operations in equalities in GA. By solving the equalities we can obtain the fold line (s) that are used in each...

We present computer-assisted construction of regular polygonal knots by origami. The construction is completed with an automated proof based on algebraic methods. Given a rectangular origami or a finite tape, of an adequate length, we can construct the simplest knot by three folds. The shape of the knot is made to be a regular pentagon if we fasten...

We present computer-assisted construction of reg-ular polygons by knot paper fold. The construction is completed with an automated proof based on algebraic methods. Given a rectangular origami or a finite tape, both of an adequate length, we can construct the simplest knot by making three folds. The shape of the knot is made to be a regular pentago...

Symbolic computation is the science of computing with symbolic objects (terms, formulae, programs, algebraic objects, geometrical objects, etc). Powerful symbolic algorithms have been developed during the past decades and have played an influential role in theorem proving, automated reasoning, software verification, model checking, rewriting, forma...

Making a knot on a rectangular origami or more generally on a tape of a finite length gives rise to a regular polygon. We present an automated algebraic proof that making two knots leads to a regular heptagon. Knot fold is regarded as a double fold operation coupled with Huzita's fold operations. We specify the construction by describing the geomet...

Tetsuo Ida Jacques Fleuriot (Eds.) Automated Deduction 9th International Workshop, ADG 2012 Edinburgh, UK, September 17-19, 2012 Revised Selected Paper

Geometrical theorem proving is challenging in many ways. One that comes to our mind is to face directly the formal-izations of basic geometrical objects together with of methods for construction and formalization of subsequent construction of the geometrical objects of study. The other is to face proving geometrical theorems formally, i.e., to foll...

Origami constructions have interesting properties that are not covered by standard euclidean geometry. Such properties have been shown with the help of computer algebra systems. Proofs performed with computer algebra systems can be accompanied by proof documents, still they lack complete mathematical rigorousity, like the one provided by proof assi...

Morley’s theorem states that for any triangle, the intersections of its adjacent angle trisectors form an equilateral triangle. The construction of Morley’s triangle by the straightedge and compass method is impossible because of the well-known impossibility result for angle trisection. However, by origami, the construction of an angle trisector is...

Mathematical foundation of origami (2D) and Computational construction and proof of origami are discussed. We presented new formalism of Huzita’s origami rules and showed that the sixth rule can simulate all the other rules.

Conference code: 84156, Export Date: 22 April 2011, Source: Scopus, Art. No.: 5715349, doi: 10.1109/SYNASC.2010.4, Language of Original Document: English, Correspondence Address: Ida, T.

Origami, i.e. paper folding, is a powerful tool for geometrical constructions. In 1989, Humiaki Huzita introduced six folding operations based on aligning one or more combinations of points and lines [6]. Jacques Justin, in his paper of the same proceedings, also presented a list of seven distinct operations [9]. His list included, without literal...

A proof document for origami theorem proving is a record of entire process of reasoning about origami construction and theorem proving. It is produced at the completion of origami theorem proving as a kind of proof certificate. It describes in detail how the whole process of an origami construction and the subsequent theorem proving are carried out...

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewrite system (O,R), where O is the set of abstract origami's and is a binary relation on O, called fold. An abstract origami is a triplet (Π,∼), where Π is a set of faces constituting an origami, and ∼ and are binary relations on Π, each repres...

This is the extended abstract of the invited tutorial on a compiler for origami construction and verification.

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...

Conference code: 80505, Export Date: 22 April 2011, Source: Scopus, Art. No.: 5460885, doi: 10.1109/SYNASC.2009.4, Language of Original Document: English, Correspondence Address: Watt, S. M.

In this talk I will show the importance of symbolic and alge- braic methods in computational origami. I discuss Huzita's axiomatization of origami and the algebraic graph rewrit- ing of abstract origami. The former is used with relation to the algorithmic treatment of origami foldability and origami geometrical theorem proving. The latter is used t...

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewrite system (O,), where O is the set of abstract origami's and is a binary relation on O, called fold. An abstract origami is a triplet (Π, , ≻), where Π is a set of faces constituting an origami, and and ≻ are binary relations on Π, each repr...

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system View the MathML source, where View the MathML source is the set of abstract origamis and right arrow-looped is a binary relation on View the MathML source, that models fold. An abstract origami is a structure (Π,reverse similar,s...

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. While Huzita's ax-ioms allow solving equations up to degree 4, it is possible to use other te...

Computational origami is the computer assisted study of mathematical and computational aspects of origami. An origami is constructed by a finite sequence of fold steps, each consisting in folding along a fold line. We base the fold methods on Huzita’s axiomatization, and show how folding an origami can be formulated by a conditional rewrite system....

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 devel- oped usi...

We present a web environment for origami theorem proving and show the experimental results for using its functionalities. The environment consists of a mathematical engine for computational origami, a graphical interface for origami construction and visualization through the web, and a system of web services for symbolic computation, which allows w...

We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewrite system (O,), where O is the set of abstract origami's and is a binary relation on O, called fold. An abstract origami is a triplet (Π,, � ), where Π is a set of faces constituting an origami, and andare binary relations on Π ,e ach repres...

We discuss some aspects of the computational origami as a new kind of science of shapes. The computational origami is closely linked to paper craft, traditional art, industrial art, geometry of all levels, and furthermore to computer science. In this talk we present models of origami for computer assisted construction of and reasoning about origami...

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 ad...

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 specif...

webOrigami2 is a system for web users to perform origami construction and to reason about geometric properties of origami. It is a web application that combines computational origami system, called Eos (E-origami system), running as a backend process; Scorum (Symbolic computation forum) which offers web services for symbolic computations; and an in...

We present a system of web services for symbolic computa-tion under development by us. It provides the means of easily accessing to symbolic computation functionalities to the users on the web by taking advantage of the resources available on networks dedicated for symbolic computations. It makes possible for the users to experiment with differ-ent...

Summary form only given. The World Wide Web is an important knowledge repository for our daily activities, and moreover it is transforming itself to service repository. Many scientists not only publish their results on the Web but offer services that accrue from their scientific discoveries and inventions. The WebOrigami, under development by SCORE...

We present an origami construction of a maximum equilateral triangle inscribed in an origami, and an automated proof of the correctness of the construction. The construction and the correctness proof are achieved by a computational origami system called Eos (E-origami system). In the construction we apply the techniques of geometrical con- straint...

Using higher-order functions is standard practice in functional programming, but most functional logic programming languages that have been described in the literature lack this feature. The natural way to deal with higher-order functions in the framework of (first-order) term rewriting is through so-called applicative term rewriting systems. In th...

Construction of geometrical objects by origami, the Japanese traditional art of paper folding, is enjoyable and intriguing. It attracted the minds of artists, mathematicians and computer scientists for many centuries. Origami will become a more rigorous, effective and enjoyable art if the origami constructions can be visualized on the computer and...

We present a computational origami construction of Morley’s triangles and automated proof of correctness of the generalized Morley’s theorem in a streamlined process of solving-computing-proving. The whole process is realized by a computational origami system being developed by us. During the computational origami construction, geometric constraint...

This paper describes the capabilities of a package for strategic programming with labeled rules. The package is called ρLog, and it extends the rule–based programming capabilities of Mathematica with: (1) new pattern matching constructs, such as context patterns and regular constraints for sequence variables and context variables, (2) a rule labeli...

We describe Huzita's origami axioms in logical and algebraic point 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. We give the logical specification of Huzita's axioms as constraints among geometric objects of origami in the l...

We describe the current capabilities of a system for rule-based programming which is being developed by us. The system is called ρLog and consists of a pattern matching system and a strategic programming system embedded into a powerful system for symbolic and numeric computation. Matching with context and sequence variables, possibly constrained by...

Prof. Eiichi Goto passed away on June 12, 2005 at the age of 74 after a long struggle with illness that was initially caused by diabetes. Despite his suffering, he continued to actively pursue his goals as a scientist, an engineer and an educator until his death. He was a Japanese pioneer in computing. His contributions in computing are so varied t...

This paper describes the capabilities of a package for strategic programming with labeled rules. The package is called ρLog, and it extends the rule–based programming capabilities of Mathematica with: (1) new pattern matching constructs, such as context patterns and regular constraints for sequence variables and context variables, (2) a rule labeli...

Symbolic and algebraic computations are one of the fastest growing areas of scientific computing. In this paper we present an overview of the state-of-the-art in symbolic and algebraic computations on parallel and distributed computers and on grids. We give some background information, including typical application areas, and then give a list of pa...

We show a stepwise origami construction of a Morley's triangle together with the automated proof of Morley's theorem and its generalization. Morley's theorem states that the three points of intersection of the adjacent interior trisectors of the angles of any triangle form an equilateral triangle. The whole process of origami construction and subse...

Origami (paper folding) has a long tradition in Japan’s culture and education. We are developing a computational origami system, based on symbolic computation system Mathematica, for performing and reasoning about origami on the computer. This system is based on the implementation of the six fundamental origami folding steps (origami axioms) formul...

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...

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 implem...

Construction of geometrical objects by origami, the Japanese traditional art of paper folding, is enjoyable and intriguing. It attracted the minds of artists, mathematicians and computer scientists for many centuries. Origami will become a more rigorous, effective and enjoyable art if the origami constructions can be visualized on the computer and...

Grid technology has emerged as a viable solution of distributed computing. In a Grid Computing Network, various heterogeneous systems are linked together, being able to interact for fulfilling some computational goal. Mathematica is a fully integrated environment for technical and scientific computing. Although there are many independent Mathematic...

We describe origami programming methodology based on constraint functional logic programming. The basic operations of origami are reduced to solving systems of equations which describe the geometric properties of paper folds. We developed two software components: one that provides primitives to construct, manipulate and visualize paper folds and th...

We describe origami programming methodology based on constraint functional logic programming. The basic operations of origami are reduced to solving systems of equations which describe the geometric properties of paper folds. We developed two software components: one that provides primitives to construct, manipulate and visualize paper folds and th...

We describe a collaborative constraint functional logic programming system for web environments. The system is called Web CFLP, and is designed to solve systems of equations in theories presented by higher-order pattern rewrite systems by collaboration among equation solving services located on the web. Web CFLP is an extension of the system Open C...

Origami is a Japanese traditional art of paper folding. Recently, Origami became a topic of active research due to its relation to art, geometry, theorem proving, and declarative programming. It has been recognized that several interesting mathematical problems can be described in an elegant way by paper folds. In this paper we describe an origami...