Simon saw a way to draw epitrochoids (gear rolling outside another gear) and hypotrochoids (gear rolling inside another gear) on VSauce: two equal circles rolling around each other form a cardiod (a heart-like shape in the Mandelbrot set), and if you take an outside circle twice as small as the inner circle, you’ll get a nefroid, if the radius of the outer circle is 1/3 of that of the inner circle, you’ll get a flower with 3 petals, if it’s 1/4 – a flower with 4 petals and so on. Basically, this is the way a spirograph works. “What if I take an irrational number?” Simon asked, all excited. The radius of the outer circle will not equal a half, or a quarter of the radius of the inner circle, but let the ratio be an irrational number. “Let’s take an easy one: 1/Phi!” Simon took his compasses and constructed the golden ratio, then subtracted 1 from it (as Phi – 1 equals 1/Phi). “I’m almost certain something beautiful is gonna pop up!”

The two circles with the ratio of 1/:

Constructing – 1:

Cutting the circles out of cardboard:

The first 1 1/4 rolls around the inner circle sort of resembled a cardioid:

Several rolls further:

And further:

Simon worked out the diameter and the circumference of the flower:

Two days later, we also tried rolling a circle in a circle (the ratio was 1/2 this time):

“It’s going to be very anticlimactic”, Simon warned.

Just a straight line!!

Simon writes: “But, the experiment wasn’t over yet. We then tried designing a handle going on to the circle:

When we cranked the handle, such that the circle rolled, we were supposed to get an ellipse, but instead of that, we got something else boring, a perfect circle (although you could say a circle is a kind of ellipse)! Then I tried it on my own, and I got a not that boring ellipse.”