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.