This month’s topics:
Soddy’s hexlet in a Japanese shrine.
During the Edo period (1603-1868) there was a custom in Japan of individuals hanging mathematical tablets in shrines or temples. On these wooden sangaku (“calculation tablets” measuring typically around 80 $\times$ 170cm) were written problems or solutions to other problems, often with elaborately colored diagrams. In a paper from the 2014 Bridges conference, Hidetoshi Fukagawa and Kazunori Horibe explain that, in the Edo era, “ordinary people enjoyed mathematics in daily life, not as a professional study but rather as an intellectual popular game and a recreational activity.” However, the careful display of sangaku in sacred precincts suggests that it was also taken quite seriously. These tablets were also surveyed by Tony Rothman in the (paywall protected) May 1998 issue of Scientific American.
A May 30, 2024 posting on Nippon.com, a site whose mission is to “share Japan with the world,” highlights three instances in which the mathematics found during the Edo period anticipated its discovery in the West, often by many years. One of these comes from a sangaku. The item in question is “Soddy’s hexlet,” named for Frederick Soddy, FRS. Soddy first announced the hexlet in verse (!) (Nature, December 5, 1936). A proof by Frank Morley was published in Nature in January of the next year. That issue also included Soddy’s elaboration of a special case of the hexlet to construct his “Bowl of Integers”.
In Soddy’s hexlet, we start with three spheres $S_1, S_2, S_3$. One of the spheres, say $S_1$, encloses both of the others. Each sphere is tangent to the other two. The statement is as follows: let $S_A$ be any sphere tangent to all three. Then $S_A$ is part of a ring of six spheres, $S_A, S_B, \dots, S_F$, which encircle $S_2$ and $S_3$ inside of $S_1$. Each sphere in the ring of six is tangent to its two neighbors in the ring, as well as to all three of $S_1, S_2, S_3$. This ring is the “hexlet.”
Morley also proved that the radii of the nine spheres are related by the identity
$$ \frac{1}{r_A} + \frac{1}{r_D} = \frac{1}{r_B} + \frac{1}{r_E} = \frac{1}{r_C} + \frac{1}{r_F} = 2\left(-\frac{1}{r_1} +\frac{1}{r_2} + \frac{1}{r_3}\right)$$
As Soddy put it:
Now these beads without flaw obey this first law
For the aggregate sum of their bends.
As each in the tunnel slims through the funnel
Its vis-à-vis grossly distends.
But the mean of the bends of each opposite pair
Is the sum of the three of the thoroughfare.
(The bend of a sphere is the inverse of its radius, except that for $S_1$, inside-out with respect to the others, the bend counts as negative.)
The sangaku in question was hung in a shrine in 1822, more than a hundred years before Soddy’s discovery, by one Irisawa Hiroatsu. Irisawa poses the problem: suppose $S_1, S_2, S_3$ and $S_A$ have diameters respectively 30, 10, 6 and 5. What are the diameters of the other spheres? He works through the calculation, obtaining for $S_B, \dots, S_F$ the diameters 15, 10, 3.75, 2.5, 2$\frac{8}{11}.$ You can check that these numbers satisfy Morley’s identity.
The original sangaku is lost, but a copy was made and appears in a book published in 1832. Details are in the report by Fukagawa and Horibe mentioned above.
AI and the future of mathematics.
Thomas Fink, the director of the London Institute for Mathematical Sciences, argues in a May 14 Nature article that conjectures are the part of math research where AI (artificial intelligence) will have the greatest impact.
In mathematics, progress comes when a mathematician has an idea (perceives a pattern, feels a correspondence, intuits a structure) and then sets out to check if that idea is correct. An idea is only dignified with the name “conjecture” if it becomes known before a proof has been established and if it has interesting implications. This is what Fink calls a “good conjecture.” Good conjectures are the way forward in mathematics, and Fink gives several examples to show how surprisingly good conjectures can be generated by AI. Nevertheless, as he states in his conclusion, for now it will take human judgment and experience to pick out the good ones: “AI will act only as a catalyst of human ingenuity, rather than a substitute for it.”
One set of examples Fink gives comes from a 2021 paper which harnessed AI to systematically look for mathematical characterizations of fundamental constants like $e$ and $\pi$. These turned out to be mostly in the form of continued fractions.
We’ll define continued fractions by example: The “golden mean” $\phi$ can be written as the continued fraction
$$\phi = 1 + \frac{1}{1+{\displaystyle\frac{1}{1+{\displaystyle \frac{1}{1+\cdots}}}}}$$
which means (enough terms are given for the pattern to be apparent) that $\phi$ is the limit of the sequence
$$1,\; 1 + \frac{1}{1+1}=\frac{3}{2},\; 1 +\frac{1}{1+{\displaystyle\frac{1}{1+1}}}=\frac{5}{3},\; 1 + \frac{1}{1+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{1+1}}}}}= \frac{8}{5},\; \cdots .$$
Among other formulas, the “Ramanujan machine” came up with
$$\frac{8}{\pi^2} = 1 – \frac{2 \times 1^4-1^3}{7-{\displaystyle\frac{2\times 2^4-2^3}{19-{\displaystyle \frac{2 \times 3^4-3^3}{37-{\displaystyle \frac{2 \times 4^4-4^3}{\cdots}}}}}}}$$
The numbers 1, 7, 19, 37 are the first hexagonal numbers. This formula, according to the authors, isn’t yet proved. Whether it is a “good” conjecture or not remains to be seen.
Fink also gives examples from the preprint Murmurations of Elliptic Curves. An elliptic curve $E$ consists of all the solutions to an equation of the form $y^2=x^3+ax+b$, just as a circle of radius 1 corresponds to the equation $x^2 + y^2 = 1$. But an elliptic curve is not only a curve corresponding to a polynomial equation. It is also, in a natural way, an abelian group (for details see here or here), and as an abelian group $E$ has a rank (analogous to the dimension of a vector space), a non-negative integer. Rank is an elusive invariant. As the “Murmurations” authors remark, “there is no general algorithm to compute the rank of an elliptic curve.” In fact the Birch and Swinnerton-Dyer conjecture, in which the rank of an elliptic curve appears, is a central open problem in the field.
In an earlier paper, the authors had used machine learning to train a computer to predict the rank of an elliptic curve. According to Lyndie Chiou, writing in Quanta, the authors were led to the discovery of murmurations by their efforts to explain why that earlier program worked so well.
Elliptic curves are important in number theory (and in cryptography). Often for these applications, we look for solutions modulo a prime $p$. That is, we want pairs of numbers $x,y$ in the range $0 \dots p-1$ where $y^2 – (x^3 + ax + b)$ is a multiple of $p$. For an elliptic curve $E$ the number of mod $p$ solutions, $N(E,p)$, is always finite. Murmurations turn up when the authors investigate the way $N(E,p)$ varies with $p$, and how this variation depends on the rank of $E$. They calculate the average value of the quantity $p + 1 – N(E,p)$ (averaged over all the elliptic curves whose coefficients satisfy certain number-theoretic inequalities) for the first 10,000 primes (from 2 to 104,729). When they plot the average values for odd-rank and even-rank curves separately, as shown in the figure below, the points form two clouds that oscillate up and down as $p$ increases (the term “murmuration” usually describes the coordinated behavior of a huge flock of starlings), with the direction of the oscillation depending on the parity of the rank. As the authors remark, this oscillation is still unexplained.
