Folding a 3D Euclidean space

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.

22 pages, 18 figures. Expanded explanation in Section 4.1 and minor corrections. This is an expanded version of the paper published in Origami7