Summary: Origami Alignments and Constructions in the
Roger C. Alperin
Neutral geometry is the geometry made possible with the first 28 theorems of
Euclid's Book 1-- those results which 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 which 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); the
area of a triangle is its defect (radian difference between and the angle sum).
As a consequence of this last remark, Bolyai (circa 1830) showed one can square
some circles (one uses a regular four sided polygon) in the hyperbolic plane using
ruler and compass, [G'04],[C'90], [J'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 which 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