Axiomatizing Origami planes

TL;DR

This paper develops a logical axiomatization of Origami geometry, connecting it to field theories, and proves that the resulting logical system is undecidable, thus formalizing the mathematical foundations of Origami constructions.

Contribution

It introduces a new axiomatization of Origami geometry based on logical axioms and links it to field theories, establishing undecidability of the system.

Findings

Axiomatization of Origami geometry using logical axioms.

Bi-interpretations between Origami theories and field theories.

Undecidability of the first-order theory of the axiomatization.

Abstract

We provide a variant of an axiomatization of elementary geometry based on logical axioms in the spirit of Huzita--Justin axioms for the Origami constructions. We isolate the fragments corresponding to natural classes of Origami constructions such as Pythagorean, Euclidean, and full Origami constructions. The sets of Origami constructible points for each of the classes of constructions provides the minimal model of the corresponding set of logical axioms. Our axiomatizations are based on Wu's axioms for orthogonal geometry and some modifications of Huzita--Justin axioms. We work out bi-interpretations between these logical theories and theories of fields as described in J.A. Makowsky (2018). Using a theorem of M. Ziegler (1982) which implies that the first order theory of Vieta fields is undecidable, we conclude that the first order theory of our axiomatization of Origami is also undecidable.