Footsteps of pi

Imagine you have a compass.

Instead of the 4 cardinal directions, mark the compass with 10 evenly-divided indications.

Let’s play a game together. You’re standing on a plane, and in front of you is a ball.

I’m going to tell you a number. Push the ball with enough force to impart a 1 meter-per-second change in velocity in the direction matching the number’s annotation on your compass.

For example, if I gave you the number 7:

Now, I’ll give you some more numbers. Rather than an absolute change in direction, you will push the already-rolling ball, adjusting its current velocity.

7, 0, 3

I’m going to give you more numbers. Many more.

What does the path of the ball look like when I tell you 580,000 digits of pi?

Addendum

I originally created this image in early 2020 to impress and woo my now-girlfriend^✱, who I adore.

The colors are arbitrary, and have no deeper meaning. I picked a simple strategy to iterate colors and liked the result aesthetically, so stuck with it.

This image would look different if you interpreted pi in a different base. There’s nothing special about base-10 in this context, it’s just what I used.

Here are some more irrational numbers expressed in this way: