Morley’s theorem revisited: Origami construction and automated proof

Abstract

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.

Formalizing polygonal knot origami

“…The method of earlier versions of Eos with concrete examples was discussed in Ida et al (2011) and Ghourabi et al (2013a). The propositions that we want to prove are of the following form:…”

Knot Fold of Regular Polygons: Computer-Assisted Construction and Verification

“…V. PROOF In this section we outline the proof method of EOS. The method with concrete examples is discussed in detail in [9] and [10]. The proposition that we want to prove is of the following form:…”

Logical and Algebraic Views of a Knot Fold of a Regular Heptagon

“…By the algebraic interpretation of the formulas we derive polynomial equalities. The problem of finding fold line(s) is therefore reduced to solving constraints expressed in multi-variate polynomials of degree 3 over the field of origami constructible numbers [8,5].…”

“…The multi-fold construction allows solving higher degree equations. We presented a construction method of angle trisection using 2-fold operation [5] and angle quintisection using 4-fold operation [10]. Although the p-fold method generates an arbitrarily high degree polynomial, accurately folding an origami by p lines simultaneously would be difficult to do by hand even for p = 2.…”

“…To obtain geometrical objects m, n, F , G, H, I and J, the formula in (7) is transformed into a set of algebraic equalities. Further details of the algebraic interpretation are explained in [8] and [5].…”