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The neuroscience of mathematical beauty, or, Equation beauty contest!

Neuroscientists Semir Zeki and John Paul Romaya have put mathematicians in an MRI scanner and shown them equations, in an attempt to discover whether mathematical beauty is comparable to the experience derived from great art.

They’ve detailed the results in a paper titled “The experience of mathematical beauty and its neural correlates”. Here’s a bit of the abstract:

We used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.

BBC News puts it: “the same emotional brain centres used to appreciate art were being activated by ‘beautiful’ maths”. This is interesting, according to the authors, because it investigates the emotional response to beauty derived from “a highly intellectual and abstract source”.

As well as the open access paper, the journal website contains a sheet of the sixty mathematical formulae used in the study. Participants were asked to rate each formula on a scale of “-5 (ugly) to +5 (beautiful)”, and then two weeks later to rate each again as simply ‘ugly’, ‘neutral’ or ‘beautiful’ while in a scanner. The results of these ratings are available in an Excel data sheet.

This free access to research data means we can add to the sum total of human knowledge, namely by presenting a roundup of the most beautiful and most ugly equations!

Most beautiful equations

The most ‘beautiful’ equation in the survey1 with 13 ‘beautiful’ votes and two ‘neutral’, was Euler’s identity:

\[ 1+e^{i \pi} = 0 \text{.} \]

A close second, with 13 ‘beautiful’ votes, one ‘neutral’ and one ‘ugly’, was Euler’s formula relating exponential and trigonometric functions:

\[ e^{ix} = \cos{x}+ i \sin{x} \text{.} \]

Honourable mentions, with twelve ‘beautiful’, two ‘neutral’ and one ‘ugly’ rating each, go to this identity for $e$:

\[ e= \lim_{n \to \infty} \left( 1+\frac{1}{n} \right)^n \text{,} \]

and Cauchy’s residue theorem:

\[ \oint _{\gamma }f(z) \,\mathrm{d}z = 2\pi i \sum \operatorname{Res}(f,a_{k}) \text{.} \]

Most ugly equation

The clear winner for most ugly, with thirteen ‘ugly’ votes (although two ‘beautiful’ as well), was this equation expressing the inverse value of $\pi$ as an infinite sum:

\[ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!(1103+26390k)}{(k!)^4396^{4k}}\text{.} \]

Most neutral equation

Only one crucial question remains to be answered: which equation was the most neutral? The most votes for ‘neutral’, eleven, went to:

\[ A \cap A^\complement = \emptyset \text{.} \]

More information

Mathematics: Why the brain sees maths as beauty at BBC News.

The paper: The experience of mathematical beauty and its neural correlates, by Semir Zeki, John Paul Romaya, Dionigi Benincasa and Michael Atiyah.

  1. Using the data taken while participants were in the scanner. []

4 Responses to “The neuroscience of mathematical beauty, or, Equation beauty contest!”

  1. Neil Bickford

    The raw data has a lot of other interesting (though likely statistically uncertain) extremes. For example, the equation the subjects were the least familiar with was Equation 42:
    \[ \chi_{\Omega }\left ( exp\, X \right ) = \int_{\Omega}e^{i\left \langle F,X \right \rangle+\sigma \left ( F \right )} \]
    the “Integral formula for a character of an irreducible representation of a Lie group corresponding to the co-adjoint orbit $ \sigma $.
    Equation 14, Ramanujan’s inverse pi formula (also the least-liked equation) was the most polarizing: every subject either ranked it beautiful or ugly.
    There’s also a table of pre-scan scores which asked the subjects to rank each equation on a scale of -5 to 5. The one with the lowest absolute value of sum of scores (another way to compute “most neutral”) was Stoke’s theorem, followed closely by Equation 42 and the continuum hypothesis.
    The “least controversial” formula was number 5 (Euler’s identity) with a variance of only 20, while the most controversial was, once again, Ramanujan’s inverse pi formula.
    Clearly, the conclusion to be made is that milliramanujans are subjectively misleading and their use should be phased out as soon as possible.
    (Oh, and one last thing: In the “identity for e” above, a should probably be changed to 1.)

    Reply
  2. Jan Van lent

    I think there may be a mistake in Equation 9 of the list.
    It says \[\mathcal{F}_x \left[e^{-a x^2}\right](k) = \sqrt{\frac{\pi}{a}} e^{-\pi^2 k^2 / a^2},\] but I think the $a^2$ should be $a$: \[\mathcal{F}_x \left[e^{-a x^2}\right](k) = \sqrt{\frac{\pi}{a}} e^{-\pi^2 k^2 / a}\]. Dimensions: $[x] = T$, $[a]=T^{-2}$, $[k]=T^{-1}$.

    Reply
  3. Jan Van lent

    There may also be a problem with Equation 35 (related to the Equation 9). The equation states \[\sqrt{\pi} \sum_{n=-\infty}^{\infty} e^{-n^2} = \sum_{n=-\infty}^{\infty} e^{-\frac{n^2}{4}},\] but this does not hold. I think it should be \[\sqrt{\pi} \sum_{n=-\infty}^{\infty} e^{-n^2} = \sum_{n=-\infty}^{\infty} e^{-\pi^2 n^2}.\]

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