Math Runes

A visualization of multiplication in various moduli

optimized for touch devices

What's going on here?

You're looking at pictures of multiplication. Every dot is a number with 0 at the top and then numbered clockwise around the circle. Like a clock. As you move the dot around the circle, you change what the multiplier is. The curves show the results of that multiplication.

Here's an example with 10 dots (modulo 10) and the number ball on 2.

This shape, or "math rune", shows what happens to every number when multiplied by 2.
There is an arc from 1 to 2, and 2 to 4, and 3 to 6, and so on. Can you see them all?

5 on the bottom goes to 10, which is the same as 0 because the circle wraps around and starts over. We call this "modulo 10". Clocks are mod 12.

Dot 6 goes to 12, which is really 2, right? Because it's mod 10, we go around.
And 7 goes to 14, which is 4.
And 8 goes to 16, which is 6, and it's really just the last digit in mod 10.
Because the last digits of the counting numbers are mod 10. Groovy.

0 goes to 0, so there is a circle around 0. If an arc comes back to itself, a circle is drawn around the dot, like the 0 and the 5 in this "times 9" rune:


(What's going on in this rune, anyways?)

These are the MULTIPLICATION RUNES, and here's a paper on a fun activity to do with these mod 10 runes.

The slider on the bottom controls how many dots there are in the clock. Here is "times 2" in modulo 30:

Experiment with the interface.
You can move the multiplier directly, or let it animate.
There are buttons for changing the direction of the animation.
You can tap in the middle to start or stop the animation.
And there's another button to change the mood.


One of the "moods" shows a circle in 12 sections, like a clock. These give fractions like 1/2, 1/3, 1/4 and 1/6, which can be interesting places to look for designs. Also the fifths are marked with black stripes.

The button in top left changes from arcs to lines to help design string art. Thanks to Glen Whitney from MoMath for the string art idea, which became a workshop at Family Fridays at MoMath.

Happy hunting!

Find out more about my artwork here.
Mike Naylor