Huzita-Justin Axioms

At the First International Meeting of Origami Science and Technology, Humiaki Huzita and Benedetto Scimemi presented a series of papers, in one of which they identified six distinctly different ways one could create a single crease by aligning one or more combinations of points and lines (i.e., existing creases) on a sheet of paper. Those six operations became known as the Huzita axioms. The Huzita axioms provided the first formal description of what types of geometric constructions were possible with origami: in a nutshell, quite a lot was possible!

Huzita Axioms
The six Huzita axioms.

The six axioms are shown to the right. It has been shown that using the six Huzita axioms, it is possible to:

  • Solve all quadratic, cubic, and quartic equations with rational coefficients;
  • Trisect an arbitrary angle;
  • Construct cube roots, including the famous problem of “doubling the cube”;
  • Construct a regular N-gon for N of the form 2i3j(2k3l+1) when the last term in parentheses is a prime (a so-called Pierpont Prime);

I won’t go into a full discussion of the Huzita axioms here, but an extensive (and excellent) discussion is to be found on Tom Hull’s website.

Huzita’s axioms formed the bedrock of the study of origami geometric constructions for many years. Thus, it came as something of a shock to learn in 2002 that (almost) everyone had missed one! Japanese folder Koshiro Hatori had, in his own investigations, found a type of single-fold alignment that could not be described in terms of any of the six Huzita axioms. In other words: there was a seventh axiom!

Hatori's Axiom
Hatori’s seventh axiom.

Hatori describes his discovery on his web site. This discovery of course raised the question: are there any more axioms out there? And the answer appears to be “no”. I have since performed a complete enumeration of all possible alignments that specify a single crease, and all feasible constructions correspond to one of the six Huzita axioms or Hatori’s seventh axiom.

As it turns out, Hatori wasn’t the first to find this axiom; in the proceedings of that very first Origami Science and Math conference, Jacques Justin published a paper, “Resolution par le pliage de l’equation du troisieme degre et applications geometriques,” in which he enumerated 7 possible combinations of alignments — which turned out to be the 6 Huzita axioms plus Hatori’s 7th. But this was in 1989! (Justin further credited Peter Messer, discoverer of the origami cube-doubling, for some part of this enumeration.) As has often happened in origami-math (and mathematics in general), independent researchers have expressed the same universal laws in the language of mathematics.

The 7 axioms have become known in some circles as as the Huzita-Hatori axioms, although the Huzita-Justin axioms is a more appropriate name, given Justin’s prior identification of all seven. At any rate, it turns out that the 7th axiom doesn’t allow the solution of any higher-order equations than the original six Huzita axioms. But it’s nice to have the complete set.

Within the mathematical theory of origami geometric constructions, the seven Huzita-Justin axioms define what is possible to construct by making sequential single creases formed by aligning combinations of points and lines. The question then arises: is it possible to solve higher-order equations by combinations of alignments that define more than one simultaneous crease? The answer is yes, and for a small taste of what is possible, check out the article on angle quintisection elsewhere in this site.

I have written up a fairly extensive discussion of geometric constructions that includes a number of origami constructions and the proof of completeness of the seven HJ axioms. You can download the pdf file below.