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.

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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. \(\Box\)

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. \(\Box\)

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. \(\Box\)

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.

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