One-, Two-, and Multi-Fold Origami Axioms

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 compass-and-straightedge constructions of Euclidean geometry can be constructed using the original 6 axioms. In fact, working independently, Martin [26] showed that the operation equivalent to Huzita’s O6 (plus the definition of a point as a crease intersection) was, by itself, sufficient for the construction of all figures constructible by the full 6 axioms and that this included all compass-and-straightedge constructions. Conversely, Auckly and Cleveland [5, 14], unaware of O6, showed that without O6, the field of numbers constructible by the other 5 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]. We note that since the other 5 of the 6 HAs can be constructed using only O6, the derived operations should perhaps be called something other than axioms. However, we will bow to 20 years of established usage and continue to call them axioms. In the same proceedings that Huzita’s original listing appeared, Justin [22] presented a list of seven distinct operations — which Justin credited, in part, to Peter Messer — including one that had been overlooked by Huzita. (A shorter list of 5 operations was also presented by Huzita and Scimemi [20].) Justin’s longer listing has been somewhat overlooked, but in 2001, Hatori [13] rediscovered Justin’s 7th operation, also shown in Figure 1. While similar to the 6 HAs, it was was not equivalent to any one of them. However, it