Roots of Unity

Mathematics: learning it, doing it, celebrating it.

These Hypocycloids Will Make You Happy

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A 2-cusped hypocycloid rolling inside a 3-cusped hypocycloid rolling inside a 4-cusped hypocycloid, all the way up to a 10-cusped hypocycloid. Or is it a web made by the coolest spider ever? Image: Greg Egan. Used with permission.

Unless you're holding a baby or a scalpel, drop everything and read this blog post about hypocycloids by John Baez. (And if you're holding a scalpel, please put away whatever device you're reading this on and pay attention to your surgery!) In addition to a lovely exposition by Baez, the post features some gorgeous animations created by Greg Egan.

Hypocycloids are relatives of the curves you make when you play with a Spirograph. On a Spirograph, you roll a circle around inside of a larger circle, with your pen on a point inside the circle, tracing out a curve called a hypotrochoid. A hypocycloid is a special kind of hypotrochoid in which the point tracing out the curve is on the edge of the circle, not the interior. (I haven't used one in decades, but I'm pretty sure you can't actually make a hypocycloid on a Spirograph because you can't put the pen all the way at the edge of the circle.)

A hypocycloid with 3 cusps, also called a deltoid, is made by rolling a circle inside a circle with a radius three times as large. Image: Sam Derbyshire at the English language Wikipedia.

You can make a hypocycloid by rolling a circle around in a circle, and you can also roll a hypocycloid around in another hypocycloid! Baez's post and Egan's illustrations explore the ways different hypocycloids can roll around inside each other. Surprisingly enough, there are connections between the pretty animations and abstract mathematical groups that turn up in theoretical physics. Just go read it!

Finally, a note to all the designers out there: I will take money out of my wallet and give it to you in exchange for a toy or piece of jewelry modeled on one of those multi-level hypocycloid things. So maybe one of you can make that happen.

The views expressed are those of the author and are not necessarily those of Scientific American.