Stars over Babylon

While planning a post about splitting fields and field extensions I ran into the problem of representing what they do visually. Typically, the field of rational numbers $\mathbb{Q}$ is the system of choice for introducing field extensions and I didn't want to stray from that too much. Thus, I started thinking of an easier problem - ways to visualize $\mathbb{Q}$.

Here's the rundown: $\mathbb{Q}$ consists of all rational numbers - numbers of the form $p/q$ where $p$ and $q$ are coprime to each other. $1, \: 9/3, \: 1/2996$ are all elements of $\mathbb{Q}$, while $\sqrt{5}, e, \pi$ are not. Taken as a set, it is countably infinite, which means that it has the "same" number of elements as it does the set of integers, $\mathbb{Z}$, as opposed to the set of real numbers $\mathbb{R}$, which is uncountably infinite. (You can read here for more info.)

Now, $\mathbb{Q}$, has an interesting topology to it: As a subspace of $\mathbb{R}$ it has points which can come arbitrarily close to a point in $\mathbb{R}$, but can never reach that point. It is a totally disconnected space - there are no "continuous" subsets of $\mathbb{Q}$, but it is also non-discrete in the sense that any open set in $\mathbb{R}$ will contain at least one element of $\mathbb{Q}$.

Take a point in $\mathbb{Q}$, for simplicity it can just be $\frac{1}{2}$, or $0.5$. How close to other elements of $\mathbb{Q}$ come to it? Let's write an array and see:

$\begin{array}{lr} x \in \mathbb{Q} & value \\ 1/2 & 0.5 \\ 1/3 & 0.333... \\ 2/5 & 0.4 \\ \vdots & \vdots \\ 260/521 & 0.49904 \end{array}$

The first few values are off by quite a bit while the last value comes close, but required some relatively large numbers in the numerator and denominator. The same holds for all elements $p/q \in \mathbb{Q}$ - the next pair $s/t$ which comes comparatively close to the value of $p/q$ will have both $s$ and $t$ be far larger than $p$ and $q$. Anyways - to the visualization part.

We can use Thomae's function as a way to visualize $\mathbb{Q}$. It is defined by:

$f(x) =\left\{ \begin{array}{l} 1/q \qquad x = p/q \\ 0 \qquad otherwise \end{array} \right.$

This function relates both the value of $p/q$ and the "size" of $p/q$. It is graphed at the top of this page. There are some pretty nice fractal properties to it: for instance, there is a "valley" under every rational point. You can see that the information of the table is captured by the graph - once the value of $1/2$ is hit then rational numbers may get arbitrarily close to it but still never reach it, and this is where these valleys come from.

This function goes by many names: the Popcorn Function, the Countable Cloud Function, and my favorite, Stars Over Babylon (hence the title).

As a parting gift, here's the reciprocal Thomae function as well as a y-axis zoom in on it. (they're too small to see in the previews, so you'll have to click on them)