Proving and Constraint Solving in Computational Origami

Electron. Proc. Theor. Comput. Sci.

Takahashi

2021

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Section: Preliminarymentioning

confidence: 99%

A New Modeling of Classical Folds in Computational Origami

Electron. Proc. Theor. Comput. Sci.

Takahashi

2021

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“…We omitted those proofs here since the proof technique is the same as what we will expound in due course. Moreover, in Ida and Buchberger (2003), it is shown that Abe's method constructs trisectors using Gröbner bases method.…”

Section: Algebraic Formulation Of Morley's Theoremmentioning

confidence: 99%

Morley’s theorem revisited: Origami construction and automated proof

Journal of Symbolic Computation

Kasem

Ghourabi

et al. 2011

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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 is impossible because of the well-known impossibility result of the angle trisection. However, by origami, the construction of an angle trisector is possible, and hence of Morley's triangle. In this paper we present a computational origami construction of Morley's triangle and automated correctness proof of the generalized Morley's theorem. During the computational origami construction, geometrical constraints in symbolic representation are generated and accumulated. Those constraints are then transformed into algebraic forms, i.e. a set of polynomials, which in turn are used to prove the correctness of the construction. The automated proof is based on the Gröbner bases method. The timings of the experiments of the Gröbner bases computations for our proofs are given. They vary greatly depending on the origami construction methods, algorithms for Gröbner bases computation, and variable orderings.

“…As part of our research in computational origami, we are developing a software system, called EOS (E-origami system) [2,3]. The system has capabilities of symbolic and numeric constraint solving, visualization of origami constructions, and assists the user in proving geometric theorems about the constructed origami by symbolic computation methods, such as of Grö bner bases and cylindrical algebraic decomposition.…”

Section: Introductionmentioning

confidence: 99%

Computational origami environment on the web

Kasem

Front. Comput. Sci. China

2008

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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.