Origami Alignments and Constructions in the Hyperbolic Plane

Abstract

Neutral geometry is the geometry made possible with the first 28 theorems of Euclid’s Book 1—those results that do not rely on the parallel postulate. Hyperbolic geometry diverges from Euclidean geometry in that there are two parallels (asymptotic) to a given line through a given point and infinitely many other lines through the point that do not meet the given line (ultraparallel). In hyperbolic geometry, similar triangles are congruent; triangles have less than 180 degrees; there are no squares (regular four sided with 90 degree angles); and the area of a triangle is its defect (radian difference between π and the angle sum). As a consequence of this last remark, Bolyai (circa 1830) showed that one can square some circles (one uses a regular four sided polygon) in the hyperbolic plane using a ruler and compass [Gray 04,Curtis 90,Jagy 95]. Our aim here is to introduce origami constructions as a new method for doing geometry in the hyperbolic plane. We discuss the use of origami for making any ruler-compass construction and the possibilities for other constructions that are not ruler-compass constructions. In the first section, we give the alignments for folding in the hyperbolic plane. These are the analogues of the classical origami axioms in the plane discussed by Huzita and Justin. There is one additional alignment that is possible. Next, we discuss the relations between the alignments in the