Modular Continued Fractions and the Snub Cube REU Project Description

Jordan Schettler

1 Motivation

This project is inspired by the following remarkable identities of Srinivasa Ramanujan:

e−2π/5 e−2π 4e−2π/3 e−2π + e−4π = 1 + = 1 + p √ −4π p √ √ −4π −8π ϕ 5 − ϕ e 6 3 − (1 + 3) e + e 1 + 1 + e−6π e−6π + e−12π 1 + 1 + 1 + ··· 1 + ···

Equation 1 Equation 2 √ Here ϕ = (1+ 5)/2 is the golden ratio. It is known that Equation 1 is intimately related to the Platonic solid with twenty faces known as the icosahedron (see Table 1); similarly, Equation 2 is related to the Platonic solid with 4 faces known as the tetrahedron.

Tetrahedron Octahedron Icosahedron

Table 1: The Regular Polyhedra (up to dual)

During the 2012-2013 academic year, I ran a undergraduate research project (see http://math. ucsb.edu/~jcs/Ramanujan/) which centered around understanding and visualizing the case of the icosahedron. Our guide was William Duke’s beautiful article Continued fractions and modular functions [Duk05]. In particular, Duke explains how Equation 1 and related identities arise from the study of a function on the complex upper half plane which is modular for the congruence subgroup Γ(5): for q = exp(2πiτ) we deﬁne

q1/5 q q2 q3 r(τ) = 1 + 1 + 1 + 1 + ···

1 The relationship of the function r(τ) to the icosahedron can be seen via the expression of the j- invariant (a fundamental modular function) as a rational function of v:

(r 20 − 228r 15 + 494r 10 + 228r 5 + 1)3 j(τ) = − r 5(r 10 + 11r 5 − 1)5

The roots of the denominator (resp. numerator) are precisely the stereographically projected vertices (resp. centers of faces) of an icosahedron inscribed in a unit sphere. See Table 2.

Inscribe Tesselate Project to C

Table 2: Stereographic Projection of Icosahedron

In fact, Equation 1 follows by recognizing that r(i) is the ﬁxed point of a rotation√ of the icosahedron. More generally, we can give an elegant Galois theoretic proof of the fact that r(i n) can be expressed in radicals over Q for every positive integer n. There is an analogous theory for Equation 2 and the tetrahedron which uses a function on the complex upper half plane which is modular for the congruence subgroup Γ(6):

q1/3 q + q2 q2 + q4 q3 + q6 s(τ) = (0.1) 1 + 1 + 1 + 1 + ··· However, the case of the tetrahedron is not as well-documented, and, in fact, Duke writes “We leave the treatment of the continued fraction [Eq. 0.1] using Γ(6) along the lines of this paper as a (chal- lenging) exercise.” Even though Ramanujan derived his identities in the early 1900s, he was both unaware of the connection to Platonic solids and of the deeper complex analytic properties. In fact, this is still very much an active research; for example, an article in 2012 [MNCB12] takes note of a new algebraic relation found for the function s(τ) above which is analogous to one previously known for r(τ). One major goal of the REU project would be to understand and visualize the case of the tetrahedron. [Note: The software we employed to do this for the icosahedron is very user friendly (Geogebra 3D and SAGE).] This will give a concrete introduction to the theory of modular functions, Riemann surfaces (perhaps even moduli spaces), and even combinatorics of the “Rogers-Ramanujan” type.

2 Further Study: The Snub Cube

Even less well documented than the tetrahedron is the possibility that some Archimedean solids may also have functions deﬁned by continued fractions which are modular up to the symmetry group. We take a rather intriguing possibility as our potential for new/interesting research. Just as the golden

2 ratio ϕ = limn→∞ Fn/Fn−1 is the limiting value of the ratio of consecutive members of the Fibonacci p3 √ p3 √ sequence (1, 1, 2, 3, 5, 8, 13, ...), the Tribonacci constant T = (1 + 19 + 3 33 + 19 − 3 33)/3 = limn→∞ Tn/Tn−1 is the limiting value of the ratio of consecutive members of the Tribonacci sequence (1, 1, 2, 4, 7, 13, 24, 44, ...). This constant is related to the (non-Platonic) solid known as the snub cube in a suggestively similar way that ϕ is related to the icosahedron. See Table 3.

s 613T + 203 Volume = 9(35T − 62)

Table 3: The Snub Cube of Edge Length 1 and the Tribonacci Constant T

On the other hand, there is a beautiful (though apparently not widely-known) continued fraction involving T which is strikingly reminiscent of Ramanujan’s Platonic identities: √ √ eπ 11/24 e−π 11 √ √ −1 = 1 − T + 1 √ e−3π 11 + e−2π 11 1 + e−π 11 − √ √ e−5π 11 + e−3π 11 1 − √ √ e−7π 11 + e−4π 11 1 − 1 − · · · This leads us to the following questions: 1. Is the a function defined by a continued fraction which is modular up to the symmetries of the snub cube which recovers the above identity? 2. If so, can we describe the situation sufficiently well enough to generate more identifies of this type?

3. Can we classify when special values can be expressed in terms of radicals over Q? 4. What’s the congruence subgroup/Riemann surface involved? 5. Does anything similar occur for other Archimedean solids? 6. What about higher dimensional polytopes?

References

[Duk05] W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 137–162. [MNCB12] M. S. Mahadeva Naika, S. Chandankumar, and K. Sushan Bairy, New identities for Ra- manujan’s cubic continued fraction, Funct. Approx. Comment. Math. 46 (2012), no. part 1, 29–44.