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Conway's Topograph Part 1

This is the first in a series of three blog posts. In the following we'll investigate a few properties of an object called Conway's topograph. John Conway conjured up a way to understand a binary quadratic form – a very important algebraic object – in a geometric context. This is by no means original work, just my interpretation of some key points from his The Sensual (Quadratic) Form that I'll need for some other posts.

Definition:

A binary quadratic form $f$ is an equation of the form:

$f(x, y) = A x^2 + H x y + B y^2.$

That is, a function of two variables which is homogeneous of degree two. The coefficients $A,$ $H,$ and $B$ and variables $x$ and $y$ are often real numbers, rational numbers or integers. $\blacksquare$

When we require the coefficients $A,$ $H,$ and $B$ as well as the variables $x, y$ to be integers, we get an integer-valued form. In his Disquisitiones Arithmeticae, Gauss asked (and largely answered) the fundamental question: what integer values can each form take? For example, you may have seen the form

$f(x, y) = x^2 + y^2,$

where it was determined that the only primes (Gaussian primes) occuring were $2$ and those odd primes congruent to 1 modulo 4.

As each form $f$ is homogenous degree two, $f(\lambda x, \lambda y) = \lambda^2 f(x, y)$. As a result, if we can understand the values of $f$ for pairs $(x, y)$ which don't share any factors, we can understand the entire set of values that $f$ takes. Also, letting $\lambda = -1,$ there is no change in the value of $f$ since $\lambda^2 = 1,$ hence it suffices to think of $v = (x, y)$ as $\pm v,$ i.e. $\left\{(x, y), (-x, -y)\right\}$.

For integers $x$ and $y,$ any point $(x, y)$ can be expressed as an integral linear combination of the vectors $e_1 = (1, 0)$ and $e_2 = (0, 1)$. So if we like, we can express all relevant inputs for $f$ in terms of two vectors. However, instead considering $e_2 = (1, 1),$ we have

$(x - y) \cdot e_1 + y \cdot e_2 = (x, y)$

and realize a different pair $e_1, e_2$ which again yield all possible integer valued vectors as integral linear combinations.

Definition:

A strict base is an ordered pair $(e_1, e_2)$ whose integral linear combinations are exactly all vectors with integer coordinates. A lax base is a set $\left\{\pm e_1, \pm e_2\right\}$ obtained from a strict base. $\blacksquare$

Definition:

A strict superbase is an ordered triple $(e_1, e_2, e_3),$ for which $e_1 + e_2 + e_3 = (0, 0)$ and $(e_1, e_2)$ is a strict base (i.e., with strict vectors), and a lax superbase is a set $\langle\pm e_1, \pm e_2, \pm e_3\rangle$ where $(e_1, e_2, e_3)$ is a strict superbase. $\blacksquare$

For our (and Conway's) purposes, it is useful to consider the lax notions and leave the strict notions as an afterthought since a binary quadratic form is unchanged given a sign change. From here forward, for a vector $v,$ we use the notation $v$ interchangeably with $\pm v$ and when referring to a base/superbase, we are referring to the lax equivalent of these notions.

Follow along to Part 2.

Update:

This material is intentionally aimed at an intermediate (think college freshman/high school senior) audience. One can go deeper with it, and I'd love to get more technical off the post.