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)

Note by Levi Adam Walker
3 weeks, 4 days ago

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Ohhh and also, It is nice to see that someone on Brilliant is writing REALLY cool notes!!!! I think this is happening after a long time..........I'm just glad!!

Aaghaz Mahajan - 3 weeks, 4 days ago

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Thank you!! I want to share my love for math with everyone :)

Levi Adam Walker - 3 weeks, 4 days ago

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Hello their, once again!! Whoah!!!! The Inverse Thomae function???? Well, the popcorn function itself is an amazing thing...... But, I never thought of defining it's inverse...........but, doesn't defining the inverse of the function, make a problem.....?? Because, supposing g(x) is the inverse of the popcorn function.............Now, g(0.5) can take ANY rational value!!! Consider this, g(1/2) = 3/2 or 5/2 or 7/2 or 11/2 or 13/2..........I mean, it isn't a function anymore right?? And also, I have never heard of the inverse popcorn function till now.........If I am wrong, could you please guide me to a link...?? Thanks :)

Aaghaz Mahajan - 3 weeks, 4 days ago

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My bad, I totally meant to say the reciprocal and not the inverse! That's what I get for writing late at night :p

Levi Adam Walker - 3 weeks, 4 days ago

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Haha X'D........no worries!!

Aaghaz Mahajan - 3 weeks, 4 days ago

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I have always taken the Stern-Brocot tree approach to visualising the rational number field over all other approaches.

A Brilliant Member - 3 weeks, 3 days ago

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Ohhh yess..........that is also a really interesting method......!! I was introduced to it from this video........!!

Aaghaz Mahajan - 3 weeks, 3 days ago

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cesar junior - 2 days, 15 hours ago

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