Folding a 3D Euclidean space

TL;DR

This paper extends origami geometry to three-dimensional space, analyzing incidence constraints and defining 47 elementary fold operations with finite solutions, advancing understanding of 3D folding geometry.

Contribution

It introduces a comprehensive set of 47 3D fold operations and analyzes their constraints and solutions, expanding origami principles into three dimensions.

Findings

Identified 47 valid 3D fold operations

Analyzed incidence constraints using reflection geometry

Explored solutions and conditions for some fold operations

Abstract

This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry of reflections. Next, a set of 3D elementary fold operations is defined, which satisfy specific combinations of constraints with a finite number of solutions. The set consists of 47 valid fold operations, and solutions to some of them are explored to determine their number and conditions of existence.