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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.

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

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





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  1. 1. iquimb 10:34 pm 12/4/2013

    This article just made me dig my old Spirograph set out of the cupboard for the first time in years. And no, you can’t achieve a full hypocycloid with a Spirograph, but you can get darned close.

    Link to this
  2. 2. romeyn5835 11:56 pm 12/4/2013

    Dear Evelyn Lamb, you can send your money to me, thank you very much. :D

    http://www.amazon.com/NPW-W4083-Hypotrochoid-Art-Set/dp/B000XJRYFW/ref=sr_1_6?ie=UTF8&qid=1386219176&sr=8-6&keywords=spirograph

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  3. 3. Dani Boy 7:22 am 12/5/2013

    Would it not be possible to cut a tiny notch out of one of the Spirograph circular holes and then use that notch as a saddle for the pen?

    You know what? I’m doing it.

    Link to this
  4. 4. Dani Boy 7:25 am 12/5/2013

    Scratch that last comment – it doesn’t work. We’ll get there eventually.

    Link to this
  5. 5. SynCallio 1:59 pm 12/5/2013

    I think you could mod a Spirograph so that this would work. You’d have to glue two gears together, a smaller one on top of a larger one. The smaller one would have a radius that was exactly the same distance as one of the holes on the larger one, which would be the hole you’d put your pen in. Then you’d have to figure out a way to support a frame above the sheet, so that there was enough clearance for the larger gear to slide underneath. So the smaller gear would be restrained by the frame, and the larger gear would keep a grip on your pen.

    At least, I think that would work. I don’t have a Spirograph to try it out with.

    Link to this
  6. 6. Last Ocean Blue 5:30 am 12/13/2013

    Notice how the cusps of each nested hypocycloid remain tangent to the next one in. Also, the area between each pair of hypocycloids remains constant, though varying in shape, as they rotate (obviously). This is the principle underlying the ingenious Gerotor pump (see Wikipedia), used in motorcycle oil pumps, among other places. I think it must be behind the Wankel engine as well.

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