Subdividing the Trefoil by Origami

In 2005, David Cox and Jerry Shurman proved that the curves they call -clovers can be subdivided into equal lengths (for certain values of ) by origami, in the cases where , 2, 3, and 4. In this paper, we expand their work to include the 6-clover. 1. Historical Background From antiquity, it was known that regular polygons with sides could be constructed with compass and (unmarked) straightedge for of one of the forms , and . In 1801, Gauss showed that the list could be expanded to include powers of two times any product of distinct Fermat primes, primes of the form . He claimed to have a proof of the converse statement, but as Pierpont noted ([1], p.79), he never actually provided it. Pierpont gives an elementary proof (i.e., without Galois theory) in his paper. In 1837, the French mathematician Pierre Wantzel resolved three celebrated ancient mathematical problems definitively, when he proved the impossibility of trisecting an arbitrary angle, duplicating the cube, or constructing a regular polygon with sides for values of other than those of Gauss using only a compass and (unmarked) straightedge. Remarkably, these same constructions can be achieved by the technique of origami (paper folding). In fact, using origami, it is also possible to trisect angles, duplicate cubes, and generally construct roots of cubic equations. This was observed by Beloch in a publication in 1936 [2]. An explication of Beloch’s work, including a survey of the history, can be found in [3]. Alternatively, with a marked straightedge, one can achieve the same result. Generalizing the notion of construction to include this or an equivalent tool and using Galois theory [4], the values of for which a regular polygon can be constructed consist of all numbers of the form where and are distinct primes of the form with , . Such primes are known as Pierpont primes. Meanwhile, Abel showed in 1828 that the lemniscate can also be divided into pieces of equal length with straightedge and compass for the same values of as for the circle. See [5] for a modern proof of this result, including the converse; see also [6]. The 2005 paper of Cox and Shurman [7] expands the family of divisible curves to include the clover. The -clover is the plane curve defined by the polar equation: where is a positive integer. This is a subfamily of the sinusoidal or sinus spirals ([8], p.194). For , the curve is the cardioid; is the circle; is the clover; is the Bernoulli lemniscate. In their paper, they prove that these first four curves can be divided into arcs of equal length by origami (paper-folding)