What the $*@&#! is the Nephroid of Freeth?

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I'll give you a hint: It has absolutely nothing to do with Star Trek, Star Wars or Dr. Who. (To my knowledge. Fanboy schooling commences anon.)

More commonly called Freeth's Nephroid (which makes it sound less like a tentacled devourer of souls and more like a little boy's pet monster), it's actually a special plane curve–which is also not as weird and confusing as it sounds. Yeah, we're talkin' about a math thing today. (This was always my "B" subject, so feel free to let me know if I'm being wrong on the Internet. Again, fanboy schooling commences anon.) Onward to knowledge…

Pictured: Not the Nephroid of Freeth. Courtesy Flickr user cole24, via CC

Plane curves are really just curved shapes that exist in a two-dimensional, as opposed to three-dimensional, space. Think circle vs. sphere. "Special" plane curves are the ones that turned out to be interesting enough that mathematicians gave them their own name. Our friend the circle is a special plane curve. Ditto the parabola (aka, the curve that's kind of shaped like a boob). And, because this is math and they just like to mess with you by making "curve" not mean what you think it means, so is the line. Yeah, the "line" is really a specific thing that the general public uses as a generic term. Like kleenex.

But back to the Nephroid of Freeth. Now we know what this thing is, in general. But why is it interesting? And what's with the name? For my own sake, I'm starting with the easy question.

Nephroids are special plane curves that are shaped sort of like kidneys. (Not has fun of an analogy as the parabola, but hey, I didn't make this one up.) They often have the appearance of something produced by a bunch of mathematicians playing with a Spiro-graph. Technically, though, the shape has to do with with the way rays of light reflect off a semicircle.

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The classic example is the pattern of shadows that you see at the bottom of a coffee cup when it's set out in the sun. Light reflects off the semicircular sides of the cup and you see nephroids on the bottom.

Freeth is a dude. Rather, was. Specifically, one T.J. Freeth, an English mathematician who died in 1904. He was pretty into strophoids, plane curves that are shaped like "a belt with a twist." They are all fancy like that.

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Strophoids are curves, but they're also things that happen to curves. Plane curves that get their panties in a bunch, if you will. So that image above is the strophoid of a line. Freeth's Nephroid, on the other hand, is the strophoid of a circle.

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See how that works?

Yes, of course you do. But you're still wondering why you care. On a practical level, I'm told that you can use the Nephroid of Freeth as a jumping off point to drawing seven-, 21-, 35-, etc.-sided shapes inside a circle. Which is great, the next time you need to do that. However, astronomer David Darling has a more practical suggestion, writing that the Nephroid of Freeth "has been described as the perfect shape for a multi-seat dining table." I'm assuming he means the kidney shape with the strophoids becoming a cutout on one side, in which case, I'd be inclined to agree.

It's also the name of a pub quiz team made up mostly of mathematicians from the University of London.

Nephroid coffee image fair use via University of Minnesota Geometry Center.

Strophoid of a line image fair use via MathCurve


Nephroid of Freeth image fair use via the Geometry Atlas.