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

Abstract

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 geometrical constraints on the fold lines to be used for the construction of a knot. The algebraic interpretation of the logical formulas allows us to solve the problem of how to find the fold operations, i.e. to find concrete fold lines. The logical and algebraic framework incorporated in a system called Eos (e-origami system) is used to simulate the knot construction as well as to prove the correctness of the construction based on algebraic proof methods.

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:…”