Subdividing the Trefoil by Origami

From antiquity, it was known that regular polygons with nn sides could be constructed with compass and (unmarked) straightedge for nn of one of the forms 2, 2 ⋅ 3, 2 ⋅ 5, and 2 ⋅ m5. 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 2 +m. 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 de�nitively, when he proved the impossibility of trisecting an arbitrary angle, duplicating the cube, or constructing a regular polygon with nn sides for values of nn 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. is 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 amarked 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 nn for which a regular polygon can be constructed consist of all numbers of the form nn m 23ppm ⋯ppnn where aa, bb a a andppm,... , ppnn are distinct primes of the form 2 3+m with uu, vv a a. Such primes are known as Pierpont primes. Meanwhile, Abel showed in 1828 that the lemniscate can also be divided into nn pieces of equal length with straightedge and compass for the same values of nn as for the circle. See [5] for a modern proof of this result, including the converse; see also [6]. e 2005 paper of Cox and Shurman [7] expands the family of divisible curves to include the clover. e mm-clover is the plane curve de�ned by the polar equation:


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 2 , 2 ⋅ 3, 2 ⋅ 5, and 2 ⋅ 5. 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 2 + . 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 de�nitively, 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. is 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 2 3 ⋯ where , and , … , are distinct primes of the form 2 3 + 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].
e 2005 paper of Cox and Shurman [7] expands the family of divisible curves to include the clover. e -clover is the plane curve de�ned by the polar equation: where is a positive integer. is is a subfamily of the sinusoidal or sinus spirals ([8], p.194). For , the curve is the cardioid; 2 is the circle; 3 is the clover; is the Bernoulli lemniscate. In their paper, they prove that these �rst four curves can be divided into arcs of equal length by origami (paper-folding) construction for certain values of , as follows. eorem 1 (see [7] ere are only �ve Fermat primes known, but more than 4000 Pierpont primes have been found; as of June 2012, the largest known Pierpont prime is 3 ⋅ 2 7033641 + 1, 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 6, the trefoil. See Figure 1. ; here is the arclength, the radial distance from the origin, and is the length of one leaf of the -clover. e function is found by inverting the arclength integral: For 1 and 2, these integrals are elementary. For 3 and 4, these are elliptic integrals, and the corresponding -clover functions are elliptic functions. For 5, the integral is no longer an elliptic integral, and the corresponding clover function is not an elliptic function. However, ( 6 2 is an elliptic function, and this turns out to be enough to prove eorem 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].

Origami Constructibility
Viewing the plane as the complex numbers ℂ, the set of points which can be constructed by origami is the smallest sub�eld 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 �ve 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 ( 3 ( lie in when ( 3 / with 0, 1, … , − 1 and 2 3 1 ⋯ where , 0 and 1 , … , are distinct Pierpont primes such that 5, 17, or ≡ 1 (mod 3 . is is the main fact we need to extend the result to the trefoil curve. If ( is in , then ′ ( is also in , since satis�es the di�erential equation: Moreover, any rational expression in ( and ′ ( is in .

The Trefoil Curve
e trefoil curve is given in polar coordinates by the equation 3 cos 3 cos 3 − 3 cos s 2 (4) e rectangular equation is us, 6 + (3 2 − 4 3 0, and of course 2 2 − 2 , from which we conclude the following.

Proposition 3. (1 is origami constructible if and only if is origami constructible.
(2 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: erefore, where 2 3 2 4 − 3 (10)

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 2 is in , then satis�es a cubic equation with coefficients in , so it is in . Recall that for the clover function, 3 ( ∫ 0 (1/ 1 − 3 . So, formula (9) shows that Now, combining Propositions 3 and 4 and formula (11) and we may parametrize the curve by (15) erefore, replacing by (2/ √ 3 in (14) and comparing with (10) and (11), we see that this is the arclength parametrization. We have shown the following.

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 1/ 3 √ 2 centered at the origin trisects each leaf of the trefoil. Construction of this circle amounts to the construction of the Delian number 3 √ 2. 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 aer Abel's untimely death), and likewise it holds for the clover and the trefoil.