Like most blogs, the Illustrated Theory of Numbers blog went on a long hiatus. Unlike most blogs, this one has returned! Really, there has not been too much to show over the past few months, but I’ve gotten back to work on Part II of the book, covering binary quadratic forms (including all discriminants, Pell-like equations, a bit on SQUFOF factorization perhaps, the class number and the “Siegel bound”). This might strike the experts as a bit out of order — where is modular arithmetic already? — but I’m sticking with “global” number theory until Part III of the book. Of course, some readers might skip Part II and go directly to Part III, but I hope they will return to Part II to learn Conway’s beautiful “topographic” approach to binary quadratic forms.
I was heavily influenced by Conway’s visual approach to binary quadratic forms, found in Chapter 1 of “The (sensual) Quadratic Form.” It’s amazing how far you can go with his topographs. I think I can go through reduction theory, symmetries (a.k.a. orthogonal groups), and finiteness of the class number, without ever needing to multiply two matrices. As recent work of Savin and Bestvina illustrates, along with recent work of my PhD student Chris Shelley, Conway’s approach generalizes in interesting ways to binary Hermitian forms and beyond.
One thing I use often in Part II is the determinant. Fortunately, for two-by-two determinants, the geometry is relatively simple. Below is a two-page spread introducing the geometric interpretation of the determinant. A helpful bonus, presented in parallel, is the discrete version: Pick’s theorem for lattice parallelograms.
From mathematical and design perspectives, I like the idea of presenting parallel proofs in visual parallel on opposite pages. The left displays a theorem in continuous Cartesian geometry; the right displays a theorem in discrete geometry. The idea of dissection is the same, but the discrete version requires a bit more care.
I’m not exactly sure what to call the theorem on the right side of the page. It certainly falls under the purview of Pick’s Theorem but really, someone must have proved it before Pick, at least for parallelograms! I wouldn’t be surprised to see it in the work of Gauss or Eisenstein, if not earlier. Unfortunately, my German is not so good (nicht so gut?), though I can recognize the frequency of “Gitterpunkt” (grid-point) and “Gitterpolygon” (polygon with vertices on grid-points) in 19th century sources. Any reader who can find an earlier reference for Pick’s theorem, even for parallelograms (rectangles don’t count on their own!) gets an acknowledgment in the book!
The utility of Pick’s Theorem is the following — it gives a cute geometric proof of the following fact well-known to algebraists: A pair and of integer vectors forms an integer basis of if and only if . Indeed, a grid-parallelogram of area one cannot cover any grid-points except its corners, by Pick’s Theorem. This avoids any mention of matrix inversion, for example.
From a design perspective, this two-page spread was a lot of fun (and a bit of work). A combination of \foreach and scoped \clip commands in TikZ allowed for the easy creation of dot-textures on the right page. Perhaps the toughest decision (and one that isn’t final) was the choice of four colors; they are called “pinkish,” “blueish”, “greenish”, and “orangeish” in my source file. Following some technical color-theory advice, I worked with a color tetrad — literally a rectangular arrangement in the HSV color wheel, converted for LaTeX via the xcolor package. Analogous regions, such as the triangles, are in the nearby colors blue and green. Less saturated colors are on the left page, where there are large regions of solid color. Fully saturated colors are on the right page, where the colors are in small dots. The real test of color will come when I print this out, along with another half-dozen copies with other rectangular tetrads of color, and see what looks the best.