She had never met her father’s father, nor wondered about him. But now she wishes she could write him a letter of thanks. She wonders if he loved Newman as much as she does. Whoever he was, he took good care of his books. Bravo, Mr. Vernon Fuller.
At night, under her cone of silence, she breezed through Newman’s first selection, “The Nature of Mathematics,” by Philip E. B. Jourdain, in two days. Since this was an overview, she nodded at the math she understood and hopped over everything she didn’t, trusting it would be elaborated on later. The second entry was “The Great Mathematicians,” by Herbert Westren Turnbull, and here was where she realized that her plan to peruse the entire series in a mere few weeks would be unworkable. On the first page there was a reference to an Egyptian priest of around 1700 BC named Ahmes who—she turns right now to that very page—“was much concerned with the reduction of fractions such as 2/(2n + 1) to a sum of fractions each of whose numerators is unity. Even with our improved notation it is a complicated matter to work through such remarkable examples as: 2/29 = 1/24 + 1/58 + 1/174 + 1/232.”
Adding fractions was a cinch, of course, she’d been able to do it since she was five. But this notion of reducing them to sums of unit fractions had never occurred to her. Indeed, a good deal of her attraction to Ahmes derived from sheer puzzlement as to why he wanted to do it in the first place. Did the Egyptians have some idea that unit fractions were more fundamental? Were they? And what would “fundamental” mean in this context?
Starting with 2/3, she assumed that reducing it to 1/3 + 1/3 was not allowed. So instead she quickly came up with 1/2 + 1/6. 2/5 and 2/7 were just as easy: 1/3 + 1/15 and 1/4 + 1/28, respectively. By the time she got to 2/9 she could see that all of these fractions could be reduced simply by subtracting the largest possible unit fraction; the difference would always be another unit fraction. 2/9 = 1/5 + 1/45. 2/11 = 1/6 + 1/66. 2/13 = 1/7 + 1/91. In each case, the quotient of the denominators of the two unit fractions was equal to the denominator of the original fraction. And if you generalized the math, you could easily see why this would always be the case. So there was a much better solution to 2/29 than what old Ahmes had come up with, namely 1/15 + 1/435. Discovering that she was smarter than the greatest of all the Egyptian mathematicians was highly gratifying. (Only later did it occur to her—maybe, for Ahmes, complexity was the point? Maybe the challenge was to string out as many unit fractions as you could? Was that more “fundamental”?)
By this time she had realized that to do Newman properly, it wouldn’t be enough to read every page, she would need to reproduce every result. Next came Thales of Miletus, 640 to 550 BC, who proved (as Mette proved again) that a circle is bisected by any diameter and that the angle inscribed in a semicircle is always right. After Thales came the Pythagoreans, a large subject. Mette proved the Pythagorean theorem three different ways. She studied triangular numbers and square numbers. Everything Turnbull mentioned led her down branching paths beyond what he himself covered. For example, she figured out all by herself that, since every odd number could be expressed as the difference between two squares, then every odd number was also the lowest term in a unique Pythagorean triple, in which the two larger terms were consecutive integers: 3-4-5; 5-12-13; 7-24-25; 9-40-41; 11-60-61; 13-84-85; etc. In this kind of Pythagorean triple, not only would the sum of the squares of the two lower terms equal the square of the largest term, but the sum of the two larger terms, unsquared, would equal the square of the smallest term. It felt like magic.
She spent a week studying the five regular solids and their properties, getting lost in calculations of their internal angles. She spent another week trying to square the circle and trisect the angle, in case somebody had missed something. She spent hours with a compass and straightedge drawing nested pentagons and pentagrams, continuing their fractal recursions down to infinitesimal points—or anyway, points smaller than a sharpened pencil point. She proved once again that the legs of a regular pentagram are golden triangles. She drew a nested recursion of a golden triangle and marveled at its beauty, the way the successively smaller triangles and golden gnomons called one another into existence like divinely matched pairs, like “turtles all the way down.” This led to the logarithmic spiral, to Robinson triangles and Sierpinski triangles, to Penrose tilings. She fooled around with Ruth-Aaron numbers, Smith numbers, Carmichael numbers. She spent several weeks filling page after page with numbers subjected to the 3n + 1 rule, graphing the results. (Mathematician Jeffrey Lagarias: “This is an extraordinarily difficult problem, completely out of reach of present day mathematics.”) She found something calming in repetitive calculations, something deeply satisfying in the slow emergence of an inexorable pattern.
She always eventually returned to Newman so that, while exploring alleyways and jungle paths, she wouldn’t miss continents. Six months after her birthday she had reached Diophantine numbers (an extremely large subject), which at page 113 out
