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# Math Finds the Best Doughnut

Posted: 04/ 2/2012 8:30 pm

After a 47-year search, mathematicians Fernando C. Marques and AndrÃ© Neves have found the best doughnut, or at least the best geometric shape for a doughnut, pictured in Figure 1, with the narrow hole only about 17 percent (3-2âˆš2) of the width of the doughnut. "Best" for mathematicians means the least bent out of shape, minimizing a physically natural elastic bending energy like certain cell membranes (defined mathematically as the integral of the mean curvature squared).

Figure 1. Mathematicians prove that the best doughnut has a small hole (top image courtesy of geom.uiuc.edu/~banchoff, thanks to Tom Banchoff, all rights reserved; bottom image courtesy of Wikipedia creative commons).

The 47-year-old puzzle is called the Willmore Conjecture. It is named after the English geometer Thomas J. Willmore.

Willmore guessed the best shape in a 1965 paper. He died February 20, 2005, seven years, almost to the day, before its solution. The solvers hail from a premier mathematics institute (IMPA) in RÃ­o de Janiero and Imperial College, London. They posted a preprint of their 96-page proof on the arXiv on Feb. 27, 2012.

A key to the proof is moving the problem from ordinary, three-dimensional space to a three-dimensional spherical universe that closes back on itself, a three-dimensional analog of the Earth's surface, where you don't have to worry about things disappearing off to infinity. There they consider the closely related problem of minimizing area, a contest in which our best doughnut is distinguished only as a sixth-class competitor (with a five-dimensional space of instabilities). The theory of such hybrid "mini-maxima" was developed by my own late thesis advisor Fred Almgren, of Princeton University, and his Ph.D. student Jon Pitts, now of Texas A&M.

Actually there is not just one optimal shape; there are infinitely many equally good shapes, all related by certain "conformal" symmetries, as in Figure 2. Such "Dupin cyclides" are being used in modeling and architecture, where they piece together to design beautiful structures. Maybe Dunkin' Donuts will be interested.

Figure 2. Mathematicians are equally happy if the hole is moved "conformally" off to one side (image courtesy of chem.ucla.edu, thanks to Xavier Michalet, all rights reserved).

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02:50 PM on 04/06/2012
This is fun math !
itruth
fideistic deist with socratic tedencies
09:26 PM on 04/05/2012
Nope,Krispy Kreme is better!
07:13 AM on 04/04/2012
best donut=lemon
JBS
Part time misanthrope & full time curmudgeon
01:07 AM on 04/04/2012
The BEST doughnut is when the sign is on that says "Hot Doughnuts Now".
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08:51 PM on 04/03/2012
Ahhh, I get the point ... the perfect toriodal donut should, logically, lead to a perfect physique for cops.
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08:49 PM on 04/03/2012
... and the point is????
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Acorn Tree
in the beginning man created god.
02:03 AM on 04/04/2012
because we could?
-swift
09:40 AM on 04/10/2012
There are no points. It's all rounded.
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Jim NLN
Hillary-Frank 2016
03:19 PM on 04/03/2012
Has anyone seen my donut? It was here a minute ago. (god this milk tastes good)
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MirageRF
03:03 PM on 04/03/2012
Donuts? Who cares?
Find me a good cruller, please.
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Cincity Cin
02:11 AM on 04/03/2012
Homer Simpson voice, HMMMMMMMM donuts..
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12:21 AM on 04/03/2012
That's no doughnut, that's a bagel:
http://www.huffingtonpost.com/2012/03/16/plain-doughnuts_n_1354275.html

Sheesh.
TakeSake
The United States for All Americans
01:40 PM on 04/03/2012
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11:59 PM on 04/04/2012
I do, too.

But so far, so good.
oneeasyrider
E=mc2: From light you exist
02:26 AM on 04/04/2012
Pretty darn clever, B. A 96-page proof debunked by a 1-page skeptical observation written by a child. Not bad. Not bad at all.
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11:36 PM on 04/04/2012
Isn't it? It takes a child to prove that mathematicians have no clothes.
LMPE
I connect the most dissimilar things
10:13 PM on 04/02/2012
Homer Simpson's gonna like this.