Facial Beauty and Fractal Geometry

What is it that makes a face beautiful Average faces obtained by photographic or digital blending are judged attractive but not optimally attractive digital exagger ations of deviations from average face blends can lead to higher attractiveness ratings My novel approach to face design however does not involve blending at all Instead the image of a female face with high ratings is composed from a fractal geometry based on rotated squares and powers of The corresponding geometric rules are more speci c than those previously used by artists such as Leonardo and D urer They yield a short algorithmic description of all facial characteristics many of which are compactly encodable with the help of simple feature detectors similar to those observed in mammalian brains This suggests that a face s beauty correlates with simplicity relative to the subjective observer s way of encoding it Each of test subjects male age yr female age yr rated the arti cial female face depicted in Figure as beautiful and as more attractive than any of the six faces in a previous study based on digital face blends The design principle is clari ed in Figure First the sides of a square were partitioned into equal intervals then certain obvious interval boundaries were connected to obtain three rotated superimposed grids thick lines in Figure with rotation angles arcsin and arcsin Higher resolution details of the fractal grids were obtained by iteratively selecting two previously generated neighbouring parallel lines and inserting a new one equidistant to both for clarity some ne grained detail is omitted in Figure Shifted copies of circles also omitted for clarity inscribable in thick lined squares of Figure scaled by powers of account for transitions between non parallel face contours such as facial sides and chin To achieve a realistic impression some colors greyscales were rst matched to those in the photograph of a real person then further edited digitally Figures isolate important feature de ning lines of the rst second and third self similar grid respectively Hundreds of alternative simple geometric designs I tried led to less satisfactory results The face satis es several ancient rules used by mathematically oriented artists such as Leonardo da Vinci and Albrecht D urer e g it is symmetrical the distance between the eyes equals one eye width or one nose width the tip of the nose is about half the way from chin to eyebrows Certain facial measurements that correlate with attractiveness are automatic by products The present feature de ning rules however are much more speci c than those used in previous work For instance they specify compare Figures and The prolongations of the left eyebrow s lower and upper edges meet the right eye lid and the right eyeshadow s upper edge respectively The upper edge of the left eye s nose near interior and the lower edge of the right eye s lie on the same line through the image center between the pupils Squares of equal size are formed by lines de ning a upper edges of left eyebrow and left nostril left side of nasal ridge and left forehead b upper edge of left eyebrow lower edge of right nostril left part of nasal ridge and In particular images herein may not be used for any commercial purposes e g as templates for computer animation or plastic surgery without explicit written consent by the author right facial boundary c upper edges of right upper lip and the left eye s nose near interior left nasal ridge and right facial boundary Rules such as above symmetrically hold for opposite face parts Figure Figure shows how large diagonal squares with low fractal resolution shape additional important features such as left and right parts of lower lip and chin and certain contours of eyes and eyebrows Figure also speci es many additional simple facial proportions based on powers of The attractiveness of the face can be explained in the spirit of a previous suggestion based on the theory of minimum description length among several patterns classi ed as comparable by some subjective observer the subjectively most beautiful is the one with the simplest shortest description given the observer s particular method for encoding and memorizing it For instance realistic representations of female faces must satisfy certain constraints concerning shape and size of de ning features Among images subjectively classi ed as being within the acceptable tolerances the ones with compact codes will be preferred Tentatively accepting this hypothesis since di erent humans with face memorizing strategies tuned by di erent subjective experiences tend to have similar preferences we are led to ask what is common among human face encoding algorithms Nobody really knows but at least it is well known that early visual processing stages of mammalian brains employ rotation sensitive edge and bar detectors According to standard information theory such detectors could contribute to an e cient code of the particular face in Figure because they allow for compressed descriptions of certain important features For example detectors that will be maximally responsive to the near vertical eyebrow contours and parallel eye and mouth contours will respond minimally to the orthogonal contours of nose and facial sides with high probability they will be either on or o Loosely speaking this corresponds to one bit of information as opposed to several bits necessary for describing intermediate values and is reminiscent of results obtained by a recent learning algorithm for arti cial neural nets that also favors codes based on features with su cient but minimal information content Similarly hypothetical detectors measuring ratios of distances between parallel lines could help to further compress the description of Figure because the same distance ratios based on powers of recur over and over More generally di erent parts of the face are encodable by reusing descriptions of other parts thus allowing for compact algorithms describing the whole the de ning features of the face have low algorithmic complexity given a description model based on simple geometric feature detectors Hence the image may be viewed as an example of low complexity art Mathematicians nd beauty in a simple proof with a short description in the formal language they are using Facial attractiveness may re ect a similar correlation between beauty and subjec tive simplicity Our results indicate that neither average faces obtained by blending nor certain attractive digital caricatures thereof are as attractive as a particular face whose essential features are compactly encodable using a simple but novel geometric construction method Future analysis of attractive face types other than the one in Figure may pro t from examining the signi cance of other especially dimensional fractal geometric patterns Generation of aesthetic faces by artists may also provide clues as to how human face recognition works