Angle Quintisection

Angle Quintisection
This figure shows the key step in performing an origami angle quintisection—division into equal fifths—by folding alone.

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. It has been mathematically proven that there are only the seven axioms, and that those folds permit the construction of solutions to arbitrary equations up to degree 4: quadratics, cubics, and quartics—but no higher.

This is a step up from the more familiar geometric constructions based on compass an unmarked straightedge, which can only solve quadratic equations. In particular, angle trisection, which requires the solution of a cubic equation, is not possible within the traditional rules of compass and unmarked straightedge. However, one must be careful to note the restriction of those rules: allowing a marked straightedge permits the solution of cubic (and therefore quartic) equations. Similarly, while the Huzita-Justin axioms (HJAs) only allow the solution of equations up to degree 4, by moving beyond the restrictions that define that set, it is possible to solve via origami equations of much higher degree.

In particular, the key characteristic of the Huzita-Justin axioms is that each describes a move in which a single crease is defined by aligning various combinations of points and lines. But it is also possible to define two, three, or more simultaneous creases by forming various alignment combinations. Some of these alignments can be broken down into sequences of HJA folds, but others, which we call non-separable, cannot be broken down into simpler units. These more complicated alignments form an entirely new class of origami “axioms”, which potentially can solve equations of considerably higher order.

Like the equation associated with angle quintisection. Dividing an arbitrary angle into five equal parts requires the solution of a fifth-order equation, and is not, in general, solvable by origami using only the Huzita-Justin axioms. (That’s for an arbitrary angle; there are certainly special cases that can be divided into fifths.)

By utilizing one or more of these more complicated axioms that define two simultaneous folds, it is possible to perform an angle quintisection, as the figure at the top of the page shows. Once a bit of precreasing is performed (all using HJA folds), four simultaneous alignments are made that define two creases; both creases must be formed simultaneously.

  • Point F is brought onto line HI;
  • The upper fold line must pass through point A;
  • Point C must lie on the upper fold line;
  • Edge FE must touch point F (after it is folded)

Once the creases are made, the upper fold line divides angle EAM (which can be arbitrary) into 2/5 and 3/5; it is a simple matter to complete the division into even fifths. (You might find it an interesting challenge to work out the geometry and prove to yourself that the division is geometrically exact.)

What good is an angle quintisection? Not much, perhaps (although I will point out that my Scorpion, opus 115 requires an angle septisection in its folding sequence). But from a mathematical point of view, it tells us that there is a larger world of origami construction beyond the Huzita-Justin axioms, waiting to be explored.

For a detailed folding sequence for the angle quintisection, pdf diagrams are given below.

Solving an angle quintisection is one example of solving an irreducible quintic equation with origami axioms. This construction uses what we call a two-fold axiom. For the 4OSME conference, Roger Alperin and I each presented papers on two-fold axioms; we then combined our work to write a paper for the conference proceedings, Origami4. You can download a copy of that paper here:

Tom Hull wrote an article for the American Mathematical Monthly giving some history of origami constructions by Margharita P. Beloch in the 1930s (the first origami solution of the general cubic) and the geometric polynomial solution method developed by Eduard Lill in the 1860s (which Roger and I made use of in the preceding).