We investigate the basic fold operations, often referred to as Huzita's axioms, which represent the standard seven operations used commonly in computational origami. We reformulate the operations by giving them precise conditions that eliminate the degenerate and incident cases. We prove that the reformulated ones yield a finite number of fold lines. Furthermore, we show how the incident cases reduce certain operations to simpler ones. We present an alternative single operation based on one of the operations without side conditions. We show how each of the reformulated operations can be realized by the alternative one. It is known that cubic equations can be solved using origami folding. We study the extension of origami by introducing fold operations that involve conic sections. We show that the new extended set of fold operations generates polynomial equations of degree up to six.
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.
Origami is the centuries-old art of folding paper, and recently, it is investigated as computer science: Given an origami with creases, the problem to determine if it can be flat after folding all creases is NP-hard. Another hundreds-old art of folding paper is a pop-up book. A model for the pop-up book design problem is given, and its computational complexity is investigated. We show that both of the opening book problem and the closing book problem are NP-hard.
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.
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