Constructions with Compass and Origami-Preliminary Version March 2011 Summary: Constructions with Compass and Origami-Preliminary Version March 2011 Roger C. Alperin 1. Introduction We describe the axioms of a single fold origami system where one may also use, in addition to origami, a compass to create circles. A circle is compass constructible iff it's center and an incident point are known. An axiom of this type was implicitly used by [ES'01], when they folded the common tangents to a circle and parabola. This allows one to construct the roots to the general quartic polynomial equation. Recently, this idea of constructions using a circle together with origami has been pursued by [KGI'11]. They added three axioms to the usual Huzita-Justin axioms for single fold origami, obtaining a system which is also not more powerful than single fold origami. However, one can perform constructions in an elegant way using these rules, as they show by implementing Archimedes method of the trisection of an angle. In this article we complete these systems by considering the full range of axioms for constructions with compass and single fold origami. There are 29 axioms in all. However although the system is not more powerful than ordinary origami, it does allow one to easily construct common tangents to Collections: Mathematics