Can beauty be quantified ?

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”.

“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.

“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 ?

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.

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.

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.

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.