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 Q\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 Q\mathbb{Q}.

Here's the rundown: Q\mathbb{Q} consists of all rational numbers - numbers of the form p/qp/q where pp and qq are coprime to each other. 1,9/3,1/29961, \: 9/3, \: 1/2996 are all elements of Q\mathbb{Q}, while 5,e,π\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, Z\mathbb{Z}, as opposed to the set of real numbers R\mathbb{R}, which is uncountably infinite. (You can read here for more info.)

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

Take a point in Q\mathbb{Q}, for simplicity it can just be 12\frac{1}{2}, or 0.50.5. How close to other elements of Q\mathbb{Q} come to it? Let's write an array and see:

xQvalue1/20.51/30.333...2/50.4260/5210.49904 \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/qQp/q \in \mathbb{Q} - the next pair s/ts/t which comes comparatively close to the value of p/qp/q will have both ss and tt be far larger than pp and qq. Anyways - to the visualization part.

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

f(x)={1/qx=p/q0otherwise 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/qp/q and the "size" of p/qp/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 1/21/2 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 Walker
10 months, 3 weeks ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

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 - 10 months, 3 weeks ago

Log in to reply

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

Levi Walker - 10 months, 3 weeks ago

Log in to reply

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 - 10 months, 3 weeks ago

Log in to reply

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 Walker - 10 months, 3 weeks ago

Log in to reply

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

Aaghaz Mahajan - 10 months, 3 weeks ago

Log in to reply

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

A Former Brilliant Member - 10 months, 3 weeks ago

Log in to reply

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

Aaghaz Mahajan - 10 months, 3 weeks ago

Log in to reply

How did you do this rendering?

Kao Cen Darach - 7 months ago

Log in to reply

Did you use the fragment shader to colour certain pixels white according to the function?

Kao Cen Darach - 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...