On the elementary single-fold operations of origami: reflections and incidence constraints on the plane

TL;DR

This paper reviews and expands the foundational axioms of origami, identifying eight elementary single-fold operations, including a previously overlooked operation of folding along a line, with formal definitions and solution conditions.

Contribution

It introduces a complete set of eight elementary origami operations, incorporating a new fold operation previously ignored, and provides formal definitions and existence conditions for all.

Findings

Identified eight elementary origami operations including a new fold operation.

Provided formal definitions and conditions for the existence of solutions.

Confirmed the completeness of the set of elementary operations.

Abstract

This article reviews the so-called "axioms" of origami (paper folding), which are elementary single-fold operations to achieve incidences between points and lines in a sheet of paper. The geometry of reflections is applied, and exhaustive analysis of all possible incidences reveals a set of eight elementary operations. The set includes the previously known seven "axioms", plus the operation of folding along a given line. This operation has been ignored in past studies because it does not create a new line. However, completeness of the set and its regular application in practical origami dictate its inclusion. Formal definitions and conditions of existence of solutions are given for all the operations.