*I've always enjoyed the abstract expressionalist works of Mark Rothko (1903-1970). However, our present era of near-perpetual copyright prevents me from showing any of his work.Instead, this is a photograph entitled, "Sun through the curtains II," by David Barrie that's in Rothko's style. There's a Flikr group with nearly a hundred thousand photos of this type.(Wikimedia Commons image.)*

One pleasing aspect of art is its symmetry, and the symmetry present in many objects of science and mathematics is also considered to be beautiful. I discussed symmetry and the concept of beauty in science in a recent article (Is Science Simply Beautiful? August 26, 2019). In 2016, the BBC did an online poll to have the public select the

*A beauty poll of twelve equations from physics and mathematics. Any physicist worth his salt should be familiar with at least ten on this list, but I must confess that I didn't know the Yang–Baxter equation until I checked Wikipedia. (Created using Gnumeric. Click for larger image.)*

It appears that the online voters knew their topic area, since the winner by far was the Dirac equation. This equation, which describes massive spin-1/2 elementary particles such as the electron, is significant for two reasons. It was the result of the first theory that combined special relativity and quantum mechanics, and it predicted antimatter in the form of the positron. The originator of this equation, Paul Dirac (1902-1984), shared the 1933 Nobel Prize in Physics with Erwin Schrödinger (1887-1961), who has another equation on this list, the Schrödinger wave equation. Although Pythagoras (c.570-c.495 BC) didn't make the list, Leonhard Euler (1707-1783) did, in the form of Euler's Identity,

This equation combines five fundamental mathematical constants, 0, 1, e, i, and π, with the fundamental mathematical operations of addition, multiplication, and exponentiation. You obtain this equation by evaluating Euler's formula,e^{iπ}+ 1 = 0

"I have never done anythinguseful. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

*Godfrey Harold Hardy (1877-1947), left, and Paul Erdős (1913-1996), right. Hardy was the mentor of number theory marvel, Srinivasa Ramanujan. Erdős was an obsessive mathematician who devoted every waking hour to mathematics. (Left image, and right image, via Wikimedia Commons and modified for artistic effect.)*

The apparent rules for beauty in a mathematical proof, as compiled on Wikipedia, are as follow:

· A proof that uses a minimum of additional assumptions or previous results.Supposedly

· A proof that is unusually succinct.

· A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or collection of theorems).

· A proof that is based on new and original insights.

· A method of proof that can be easily generalized to solve a family of similar problems.

· The similarity of simple mathematical arguments to landscape paintings.In the first study, individuals were asked to match four mathematics proofs to four landscape paintings based on how aesthetically similar they found them. The proofs were the sum of an infinite geometric series, a method for summing a sequential list of positive integers attributed to a youthful Carl Friedrich Gauss, the pigeonhole principle, and a geometric proof of a Faulhaber formula.[6] The landscape paintings were "Looking Down Yosemite Valley, California" by Albert Bierstadt, "A Storm in the Rocky Mountains, Mt. Rosalie" by Albert Bierstadt, "The Hay Wain" by John Constable, and "The Heart of the Andes" by Frederic Edwin Church.[6]

· The similarity of simple mathematical arguments to pieces of classical piano music.

· The commonality of reasons for the feeling of beauty for artworks and mathematical arguments.

*A geometric determination of the sum of an infinite series, one of the four mathematical arguments used in the study.(Created using Inkscape. Click for larger image.)*

For the second study, the mathematical proofs were the same, and the classical piano music pieces were Franz Schubert's "Moment Musical No. 4, D 780 (Op. 94)," Johann Sebastian Bach's "Fugue from Toccata in E Minor (BWV 914)," Ludwig van Beethoven's "Diabelli Variations (Op. 120)," and Dmitri Shostakovich's "Prelude in D-flat major (Op.87 No. 15)."[6] The third study asked another group of participants to rate, on a scale of zero to ten, the beauty of each of the four artworks and mathematical arguments as well as scoring them in the nine categories listed above.[6] The overall results demonstrated a considerable consensus among average people in the comparison of math and art, and somewhat less of a consensus in the judgment of classical piano music and mathematics.[7] As Samuel G.B. Johnson, a study author and a Lecturer at the the University of Bath School of Management, summarizes,

"Laypeople not only had similar intuitions about the beauty of math as they did about the beauty of art, but also had similar intuitions about beauty as each other. In other words, there was consensus about what makes something beautiful, regardless of modality...There might be opportunities to make the more abstract, more formal aspects of mathematics more accessible and more exciting to students... and that might be useful in terms of encouraging more people to enter the field of mathematics."[7]

- Melissa Hogenboom, "You decide: What is the most beautiful equation?" BBC, January 20, 2016.
- Massimo Pigliucci, "Richard Feynman was wrong about beauty and truth in science," Aeon, June 28, 2019.
- Sabine Hossenfelder, "Beauty is truth, truth is beauty, and other lies of physics," Aeon, July 11, 2018.
- Sabine Hossenfelder, "Lost in Math: How Beauty Leads Physics Astray," Basic Books (June 12, 2018), 304 pp., ISBN-13: 978-0465094257 (via Amazon).
- Samuel G.B. Johnson and Stefan Steinerberger, "Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas," Cognition, vol. 189 (August 2019), pp. 242-259, https://doi.org/10.1016/j.cognition.2019.04.008.
- Kendall Teare, "Study shows we like our math like we like our art: beautiful," Yale University Press Release, August 7, 2019.
- People can see beauty in complex mathematics, study shows, University of Bath Press Release, September 5, 2019.