Let us all percolate together

I am indebted to Terence Tao for this highly comprehensible Executive Summary:

When one heats an iron bar magnet above a certain special temperature — the Curie temperature — the iron bar will cease to be magnetised; when one cools the bar again below this temperature, the bar can once again spontaneously magnetise in the presence of an external magnetic field. This phenomenon is still not perfectly understood; for instance, it is difficult to predict the Curie temperature precisely from the fundamental laws of physics, although one can at least prove that this temperature exists. However, Chayes, McKellar, and Winn have shown that for a certain simplified model for magnetism (known as the Ashkin-Teller model), the Curie temperature is equal to the critical temperature below which percolation can occur; this means that even when the bar is unmagnetised, enough of the iron atoms in the bar spin in the same direction that they can create a connected path from one end of the bar to another. Percolation in the Ashkin-Teller model is not fully understood either, but it is a simpler phenomenon to deal with than spontaneous magnetisation, and so this result represents an advance in our understanding of how the latter phenomenon works.

The advantage of the Ashkin-Teller model is that its one-dimensional nature makes it possible to solve exactly, even if its correspondence to reality is occasionally a trifle askew.

The Chayes-McKellar-Winn paper is available in PDF form from McKellar’s Web site, and this is the McKellar in question:

Danica McKellar circa 2009

I do not claim to understand all the mathematics involved. I am delighted, however, that she does: by now Danica McKellar might be better known for her several math books than for playing Winnie Cooper in the not-quite-forgotten TV series The Wonder Years. (She’s thirty-seven years old today.)


  1. Luther »

    3 January 2012 · 9:45 pm

    That’s a great story, not, neccesarily, the theoretical framework of percolation, but of McKellar’s life.

    Thanks for bringing it up.

  2. CGHill »

    3 January 2012 · 10:14 pm

    I read through that entire paper; I’m guesstimating I understood somewhere around 15 percent of it at most.

  3. Tatyana »

    4 January 2012 · 7:47 am

    Can I ask you what is the point of this post?
    You didn’t know that some of the scientists are women? You didn’t know that women have bodies? You didn’t know that some women-scientists write papers in collaboration with their co-workers?

    I won’t pretend I want to read that paper or attempt to understand what’s that about (that is only fair -I am sure McKellar will be bored to tears if I talked MY shop). What I see on the photo is a woman who is very uncomfortable. Her posture is strained, the gesture of her hands is defensive, and what about her forced smile!
    I am not going to comment on her dress, but she would do better if she covered her disproportional forehead with bangs and coordinated her earrings with her pendant.

  4. CGHill »

    4 January 2012 · 8:06 am

    It’s the inevitable result of wishing to boost traffic levels on weekends (per Rule 5), while not wanting to just throw up a bunch of pictures and say “Isn’t she pretty?”

    Were I to do a piece on Hedy Lamarr, there would be a spiffy picture, yes — but there would also be mention of her innovations in spread-spectrum communication technology, something that is used in every CDMA phone to this day.

  5. Tatyana »

    4 January 2012 · 8:24 am

    Lamarr’s innovation had practical consequences – and she was a star, a real star, not a “voice actress”.

    Other than collaborating on that paper in college, McKellar did not pursue mathematical or – wider – scientific career. She is not a scientist; if you wanted a poster girl for science, she would be a bad choice – as she continued playing minor roles in TV series* and such instead of laboring in academia or turning to utilitarian application for math in business career. For someone with as brilliant mathematical abilities as she is reported to have she basically wasted her talent; you can’t seriously call 3 pop-math books for children an achievement for someone with her potential.

    As for “pretty”….well, she is not, quite simply. She has a sporting lean figure -yes, but her face is disproportional with heavy chin (a J. Leno female lookalike!) and huge forehead.

    Compare her, f.i., with her sister, the corporate lawyer. Or with her collaborator on that paper, Brandy: ” B., incidentally, was the only student in my topology class who did better than Danica; she has recently obtained a PhD in mathematics from U. Chicago, with a thesis in PDE.”.
    This is an achievement.
    Not playing an anonymous girl Ted found in his bed one morning in 1 (one) episode of “How I met your mother”.

    And she named her son Drago.

  6. McGehee »

    4 January 2012 · 8:28 am

    by now Danica McKellar might be better known for her several math books than for playing Winnie Cooper in the not-quite-forgotten TV series The Wonder Years.

    Among those who never saw the TV series, maybe.

    I for one still take notice when she shows up as a guest on a show I watch. I regretted her character’s death on an episode of “NCIS” more than I should have, I think.

  7. Tatyana »

    4 January 2012 · 8:45 am

    Kevin, did you click on that link to “math books”? Would you characterize them as MATH books?

    Some child actors should leave their acting career after they grow to be adults. Especially if they have brilliant rare math talent.

  8. McGehee »

    4 January 2012 · 2:00 pm

    Tat, I’m male. Deal with it.

  9. Tatyana »

    4 January 2012 · 2:58 pm

    Why would I have to deal with YOUR problem?

  10. Luther »

    4 January 2012 · 8:22 pm

    This is America. People do wild and crazy things. Hardly ever according to script.

    Anyone, who can have even a small influence on children discovering mathematics for the pure joy of it, no matter the length of those child’s interest, has done something good.

    Yeah, I don’t know how to write. Sue me.

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