 
Summary: Constructions with Compass and
OrigamiPreliminary 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 HuzitaJustin 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
