ABSTRACT
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 [Alperin and Lang 09]. There is one additional alignment that is possible. Next, we discuss the relations between
the alignments in the context of neutral geometry, Euclidean geometry, and hyperbolic geometry. This leads up to the the relation of these alignments to the ruler-compass constructions in the hyperbolic plane. Basically, we show that one can do ruler-compass constructions equivalently with a subset of the folds, similar to origami in the Euclidean plane.
