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Can beauty be quantified?

Samy Lakhal
Samy Lakhal
PhD student of the Econophysics & Complex Systems Chair (LadHyX) at École Polytechnique (IP Paris)
Michael Benzaquen
Michael Benzaquen
Researcher at INSIS-CNRS and holder of the “Econophysics & Complex Systems” chair at École Polytechnique (IP Paris)

Why does there seem to be a con­sen­sus in appre­ci­a­tion of cer­tain images? What under­ly­ing prop­er­ties and rules deter­mine a pref­er­ence of beau­ty? Physi­cists from École Poly­tech­nique, in col­lab­o­ra­tion with the Art­In­Re­search gallery, pro­vide ele­ments of an answer in a recent pub­li­ca­tion1 in a field at the cross­roads of dis­ci­plines: “quan­ti­ta­tive aesthetics”.

A universalist vision of beauty.

“Beau­ti­ful is what, with­out a con­cept, is liked uni­ver­sal­ly.” Emmanuel Kant2

The con­cept of beau­ty, con­sid­ered as a uni­ver­sal and intrin­sic prop­er­ty of an object, appears in many artis­tic con­tri­bu­tions, them­selves root­ed in all peri­ods of his­to­ry. From ancient Greece to the Renais­sance, from archi­tec­ture to paint­ing, this quest for the ide­al has often con­verged towards the estab­lish­ment of pre­cise rules of rep­re­sen­ta­tion. These rules, some­times very quan­ti­ta­tive, can mate­ri­alise in dif­fer­ent ways with­in works of art: pro­por­tions of bod­ies for sculp­tures; act dis­tri­b­u­tions and dia­logue in the­atre; sym­me­try in archi­tec­tures, and many oth­er exam­ples. Even for con­tem­po­rary works that are in line with these his­tor­i­cal dis­ci­plines, these rules remain and con­sti­tute the first things that are taught on the aca­d­e­m­ic level.

Order and chaos

“Total chaos is dis­qui­et­ing. Too much reg­u­lar­i­ty is bor­ing. Aes­thet­ics is per­haps the ter­ri­to­ry in-between.” Jean- Philippe Bouchaud3

Yet, one can­not help but think of those out­stand­ing works that present at least a sen­si­tive trans­gres­sion of these rules. How many acts and sets are there in Edmond Rostand’s Cyra­no de Berg­er­ac? What lev­el of de-struc­tur­ing is there in Picasso’s Guer­ni­ca? In a way, isn’t the pow­er­ful work the one that trans­ports us between rules and their rejec­tion, the known and the unknown, order and chaos?

What could be more telling for researchers in sta­tis­ti­cal physics, where con­cepts like sud­den sym­me­try break­ing, crit­i­cal­i­ty and insta­bil­i­ty are at the heart of the discipline?

Entropy as a measure of disorder.

From the point of view of physics, order and dis­or­der are famil­iar notions. A mag­net con­sists of an almost infi­nite num­ber of micro-mag­nets, all aligned and ordered. But if the tem­per­a­ture is increased above the “Curie tem­per­a­ture”, this order is bro­ken, the micro-mag­nets become inde­pen­dent, and the over­all mag­neti­sa­tion is lost. The mea­sure of this dis­or­der and break­ing of sym­me­try is entropy, a fun­da­men­tal quan­ti­ty in ther­mo­dy­nam­ics. More for­mal­ly, entropy is the num­ber of ways to organ­ise a sys­tem so that it keeps the same phys­i­cal prop­er­ties. There are many ways to cre­ate dis­or­der, but few ways to build order.

A simple experiment on visual preference

Dur­ing our research, we asked the fol­low­ing ques­tion: is there an entropy val­ue for class­es of sim­ple abstract images that would max­imise their aes­thet­ic qualities?

To answer this ques­tion, we first gen­er­at­ed two class­es of abstract images well dis­trib­uted over three known mea­sures of dis­or­der: frac­tal dimen­sion; algo­rith­mic com­press­ibil­i­ty; and pat­tern size dis­tri­b­u­tion with­in an image. Each of these class­es of images tran­si­tions from ordered (left) to dis­or­dered (right) states. We then per­formed sur­vey exper­i­ments to deter­mine which images were most appre­ci­at­ed, first with the help of our col­leagues (Lad­HyX, CFM, ENSAE), and then through adapt­ed dis­tri­b­u­tion plat­forms (Ama­zon MkTurk). In total, near­ly 1000 peo­ple par­tic­i­pat­ed in the dif­fer­ent exper­i­ments. The results con­firmed our intu­ition: the images (a4,b4) obtain the best scores.

Entropy and structural complexity

Intu­itive­ly, extreme entropy val­ues tend to bore us or lose us. Con­verse­ly, inter­me­di­ate val­ues cap­ture our atten­tion by max­imis­ing the pres­ence of inter­est­ing struc­tures – in the form of intel­li­gi­ble pat­terns that make the image unique. In a way, our brain recog­nis­es shapes and pat­terns while “eras­ing” unnec­es­sary noise. To sim­u­late this mech­a­nism, we then pro­posed a mea­sure of “struc­tur­al com­plex­i­ty” (in red on graph above) high­light­ing these struc­tures. In doing so, we obtained a sec­ond result: the images of inter­me­di­ate entropy present, indeed, the great­est struc­tur­al com­plex­i­ty. It should be not­ed that nat­ur­al images, such as pho­tographs of forests or land­scapes, present an entropy (sta­tis­ti­cal­ly con­densed) around this same inter­me­di­ate value.

A contribution in a growing field: quantitative aesthetics.

Today, cam­eras, retouch­ing soft­ware and even search engines all ben­e­fit from recent con­tri­bu­tions in quan­ti­ta­tive aes­thet­ics45. This dis­ci­pline, by com­bin­ing mas­sive data banks6, image pro­cess­ing meth­ods and advanced learn­ing archi­tec­tures7, has sig­nif­i­cant­ly improved the auto­mat­ic scor­ing and clas­si­fi­ca­tion of images. How­ev­er, despite the per­for­mance of these tools, we can regret their inabil­i­ty to pro­vide ele­ments of under­stand­ing on the real mech­a­nisms of appre­ci­a­tion. They are, in a way, automa­ta whose rules escape us.

Our work, even if it is part of the same pre­dic­tive approach, aims at putting the inter­pretabil­i­ty of the results at the cen­tre. A fun­da­men­tal­ly phys­i­cal approach in short.

1Lakhal, S., Dar­mon, A., Bouchaud, J.-P., & Ben­za­quen, M. (2020). Beau­ty and struc­tur­al com­plex­i­ty. Phys­i­cal Review Research, 2(2), 022058
2I. Kant, Cri­tique of the Pow­er of Judg­ment
3J.-P. Bouchaud, Leonar­do 41, 239 (2008)
4H. Maître, Essai sur l’esthétique en pho­togra­phie numérique , p. 171
5H. Maître, Juger du Beau avec sub­jec­tiv­ité : le défi de l’esthétique com­pu­ta­tion­nelle, ArtSci
6C. Kang, G. Valen­zise, et F. Dufaux, EVA: An Explain­able Visu­al Aes­thet­ics Dataset, in Joint Work­shop on Aes­thet­ic and Tech­ni­cal Qual­i­ty Assess­ment of Mul­ti­me­dia and Media Ana­lyt­ics for Soci­etal Trends
7J. McCor­ma­ck et A. Lomas, Deep learn­ing of indi­vid­ual aes­thet­ics », Neur­al Com­put & Applic, 2021


Samy Lakhal

Samy Lakhal

PhD student of the Econophysics & Complex Systems Chair (LadHyX) at École Polytechnique (IP Paris)

Samy Lakhal's research focuses on applications of statistical physics to quantitative aesthetics and fracture mechanics. His work is supervised by Professors Michael Benzaquen and Jean-Philippe Bouchaud of the Econophysics & Complex Systems Chair at École Polytechnique, and by Professor Laurent Ponson of the Institut Jean le Rond d'Alembert at Sorbonne University.

Michael Benzaquen

Michael Benzaquen

Researcher at INSIS-CNRS and holder of the “Econophysics & Complex Systems” chair at École Polytechnique (IP Paris)

After working in the financial industry at Capital Fund Management (CFM), Michael Benzaquen joined the CNRS and the Hydrodynamics Laboratory of Ecole Polytechnique (LadHyX). He quickly built up a research team around themes related to Econophysics. He is a lecturer in the Economics Department of the Institut Polytechnique and teaches financial markets and the physics of social sciences at ENSAE. In 2018 he founded the X-CFM Chair “Econophysics and Complex Systems” marking the arrival of a new discipline on the Campus.