Geometric and arithmetic relations concerning origami

TL;DR

This paper formalizes geometric instruments and tools in origami, distinguishing geometric and arithmetic aspects, and explores their relations with other instruments through a new axiomatic approach.

Contribution

It introduces a formal framework for geometric instruments in origami, combining geometric and arithmetic perspectives, and reinterprets known results with new relations to other instruments.

Findings

Reformulation of known origami results

Introduction of a formal tool and map concept

New relations between origami and other geometric instruments

Abstract

We present a formalization of geometric instruments that considers separately geometric and arithmetic aspects of them. We introduce the concept of tool, which formalizes a physical instrument as a set of axioms representing its geometric capabilities. We also define a map as a tool together with a set of points and curves as an initial reference. We rewrite known results using this new approach and give new relations between origami and other instruments, some obtained considering them as tools and others considering them as maps.