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Geometry
Volume 2013 (2013), Article ID 897320, 4 pages
http://dx.doi.org/10.1155/2013/897320
Research Article

Subdividing the Trefoil by Origami

Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA

Received 7 November 2012; Accepted 21 November 2012

Academic Editor: Michel Planat

Copyright © 2013 Joel C. Langer and David A. Singer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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) construction for certain values of , as follows.

Theorem 1 (see [7]). For any , the cardioid can be divided into arcs of equal length by straightedge and compass. The circle can be divided into equal lengths by origami if and only if where , and are distinct Pierpont primes. In the case of the lemniscate, the Pierpont primes must satisfy or . The clover can be divided into equal lengths by origami if and only if where , and are distinct Pierpont primes such that , , or .

There are only five Fermat primes known, but more than 4000 Pierpont primes have been found; as of June 2012, the largest known Pierpont prime is , which has 2117338 digits. ([9]; 16th on the list of largest primes.) See [10] for up-to-date information.

In this paper, we observe that these results on origami construction can be extended to the case , the trefoil. See Figure 1.

fig1
Figure 1: (a) The clover. (b) The trefoil.

Theorem 2. The trefoil can be divided into equal lengths by origami if and only if where , and are distinct Pierpont primes such that , , or .

Cox and Shurman define the m-clover function by ; here is the arclength, the radial distance from the origin, and is the length of one leaf of the -clover. The function is found by inverting the arclength integral: For and , these integrals are elementary. For and , these are elliptic integrals, and the corresponding -clover functions are elliptic functions. For , the integral is no longer an elliptic integral, and the corresponding clover function is not an elliptic function. However, is an elliptic function, and this turns out to be enough to prove Theorem 2.

Note. Practically all of the hard work can be found in [7], to which we refer the reader for the details of our arguments. For detailed information about Galois theory, especially its application to subdividing the lemniscate, see [4]. A discussion of origami numbers can be found in [11].

2. Origami Constructibility

Viewing the plane as the complex numbers , the set of points which can be constructed by origami is the smallest subfield containing the rational numbers and closed under rational operations and under square roots and cube roots. If is a root of a polynomial of degree less than five with coefficients in , then itself is in . To subdivide a leaf of the trefoil into equal lengths, it suffices to show that the and coordinates of the division points are numbers in .

In their proof of the clover theorem, Cox and Shurman show that the values of the clover function lie in when with and where , and are distinct Pierpont primes such that , , or . This is the main fact we need to extend the result to the trefoil curve. If is in , then is also in , since satisfies the differential equation: Moreover, any rational expression in and is in .

3. The Trefoil Curve

The trefoil curve is given in polar coordinates by the equation The rectangular equation is

Thus, , and of course , from which we conclude the following.

Proposition 3. is origami constructible if and only if is origami constructible.
is origami constructible implies that is origami constructible.

From the polar coordinate formula, it is easy to derive the arclength formula in terms of ([7], p.686): Now, if we make the miraculous substitution then the following equations hold: Therefore, where

Proposition 4. is origami constructible if and only if is origami constructible.

Proof. If is in , then is the square root of an element of and, therefore, an element of . Conversely, if is in , then satisfies a cubic equation with coefficients in , so it is in .

Recall that for the clover function, . So, formula (9) shows that Now, combining Propositions 3 and 4 and formula (11) with Cox and Shurman’s result about yields the proof of Theorem 2.

The arclength parametrization of the curve can be derived using lines through the origin.

Letting , where is a parameter, we have and we may parametrize the curve by

If we then replace by , we have

Now, observe that Therefore, replacing by in (14) and comparing with (10) and (11), we see that this is the arclength parametrization. We have shown the following.

Theorem 5. The coordinates of the arclength parametrization of the trefoil are rational expressions in the elliptic functions and .

4. Concluding Remarks

As is the case for the clover, also a leaf of the trefoil cannot be subdivided into, for example, three equal arcs using straightedge and compass. In fact, it can be shown that the circle of radius centered at the origin trisects each leaf of the trefoil. Construction of this circle amounts to the construction of the Delian number . In other words, the problem of trisecting one leaf of the trefoil is equivalent to the classical problem of duplication of the cube!

It is also worthwhile to note that the algebraic problem of division into equal pieces by radicals (i.e., solvability) was achieved for the circle by Gauss, for the lemniscate by Abel (with the help of Liouville after Abel’s untimely death), and likewise it holds for the clover and the trefoil.

References

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