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What makes an equation beautiful?

By James Dacey

Earlier this year I wrote about a psychology experiment that revealed that mathematicians appreciate beautiful equations in the same way that people experience great works of art. In the experiment, which conjures up a slightly comical scene, mathematicians were hooked up to a functional magnetic resonance imaging (fMRI) machine and asked to view a series of equations. When the subjects looked at equations they had previously rated as beautiful, it triggered activity in a part of the emotional brain associated with the experience of visual and musical beauty. The formula most commonly rated as beautiful in the study, in both the initial survey and the brain scan, was Euler’s equation, eiπ+ 1 = 0.

Inspired by this study, we have put together this infographic to dissect the Euler identity and try to understand why so many mathematicians are enamoured with this little equation. Let us know what you think of the infographic and what you think are the most beautiful equations. Either post a comment below this article, or let us know on Twitter using the hashtag #BeautifulEquations.

Infographic about beautiful equations

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  1. M. Asghar

    The De Broglie equation:
    λ = h/p,
    h: the Planck constant; p, the momentum of the particle and λ, its associated wavelenght. This equation led to the “particle- wave dualty concept” and exraordinary idea of “wavefunction” and the Schrodinger’s equation.

    • Isidre

      But this is a definition rather than an equation. De Broglie’s gives sense to a particle’s wavelength, while in Euler’s all terms have been previously defined in other ways. I mean, no one says “-What is 1? +1 is -e^i*π”, what is actually true.

  2. Paul

    I like the Infographic, but you’ve reversed the captions for the basic operators. FYI. :)

  3. Trackback: Mathematicians appreciate beauty in equations | art and math

  4. Tim

    And this whole time I thought I had addition “+” and multiplication “x” backwards!

  5. StBob

    I prefer to think of it as saying that e^pi is the ith root of -1.

  6. James Dacey (article author)

    Thank you for pointing out that we had the addition “+” and multiplication “x” symbols in the wrong order. This article now contains an updated version of the infographic.

  7. Although it is a mite complicated, I find the Lorentz Transform equations to be gorgeous considering how much they do with so little.

  8. I like 6^3 – 5^3 – 4^3 – 3^3 = 0 similar to 5^2 – 4^2 – 3^2 = 0, and the cyclic number 0.142857

  9. What a beautiful topic! However, I am missing Einstein’s field equations of gravitational interactions, or have I overlooked it?

  10. george ellis

    Great post, but you missed out the crucial importance of the “=” sign. Until we had that, we had no equations

  11. Ah, my favourite equation. I think an equation is beautiful when it is simple and yet connected to many different ideas.

  12. Chris

    I like the Euler-Lagrange equation from variational calculus. So simple yet powerful.

  13. Trackback: Physics Viewpoint | What makes an equation beautiful?

  14. Dominic Parnell

    I quite like the equation for wave-particle duality concerning the speed of light constant c
    c = 1/square root(permeability x permittivity of free space).

  15. Trackback: Blog – – E Science


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