 
Summary: One, Two, and MultiFold Origami Axioms
Roger C. Alperin and Robert J. Lang
December 6, 2006
1 Introduction
In 1989 [15], Huzita introduced the six origami operations that have now
become know as the Huzita Axioms (HAs). The HAs, shown in Figure 1,
constitute six distinct ways of defining a single fold by bringing together
combinations of preexisting points (e.g., crease intersections) and preexisting
lines (creases and/or the fold line itself).
It has been shown that all of the standard compassandstraightedge con
structions of Euclidean geometry can be constructed using the original 6 ax
ioms. In fact, working independently, Martin [25] showed that the operation
equivalent to Huzita's O6 (plus the definition of a point as a crease intersec
tion) was, by itself, sufficient for the construction of all figures constructible
by the full 6 axioms and that this included all compassandstraightedge
constructions. Conversely, Auckly and Cleveland [5, 14], unaware of O5, O6,
showed that without O5, O6, the field of numbers constructible by the other
4 HAs was smaller than the field of numbers constructible by compass and
straightedge. An analysis of the hierarchy of fields which can be constructed
using different axioms systems is detailed in [2], [4].
