Beauty and the Golden Ratio

Advances in Social Sciences Research Journal – Vol. 9, No. 9 Publication Date: September 25, 2022 DOI:10.14738/assrj.99.13088. Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. Beauty and the Golden Ratio Luis F. Copertari Computer Engineering Program Autonomous University of Zacatecas (UAZ), Zacatecas, México Gloria V. Reyna Psychology Department Autonomous University of Zacatecas (UAZ), Zacatecas, México ABSTRACT The ideas of beauty and, specifically, female facial beauty, are developed in depth in this paper, in light of the adjustment of facial beauty to the golden ratio according to the parameters defined for such purpose. An experimental design of our own is made, software for carrying out the experiments is developed and the experimental results are analyzed for both male and female population from the Computer Engineering Program of the Autonomous University of Zacatecas (UAZ) and the Psychology Department, also at UAZ. It is concluded that women are more sensitive than men to the aesthetics dictated by the conformance to the golden ratio. Also, we conclude that the golden ratio does play an important role in female facial beauty perception. Keywords: Beauty, golden ratio, facial, gender, aesthetics, interface. INTRODUCTION This research originated from the curiosity derived from a video by Alexs Syntek published in YouTube entitled “ASOMBROSA-MENTE. Las Reglas de la Atracción” (Syntek, 2020). In this video, the idea of the golden ratio as a way to measure the attractiveness of the faces of female models, as well as other issues related to female and male attractiveness are discussed. These ideas are developed in depth in this research, our own experiments are carried out and a thorough theoretical revision is made both concerning the golden ratio and issues related to psychology as well as probability and statistics issues in order to analyze the experimental results obtained from students from the Computer Engineering Program of the Autonomous University of Zacatecas (UAZ) and the Psychology Department also at UAZ. In order to carry out the experiments, a friendly user interface was designed offering the experimental subject the possibility of ordering the female model according to their beauty by interacting with the software while the software counts automatically the time the user takes in doing the test. At the moment of finishing the experiment, the software adds the new statistics in the database being created in the same folder where the software is located. The software records the time of the experiment (in seconds), as well as a numerical score indicating how close to the golden ratio metric the student is. Services for Science and Education – United Kingdom Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Before carrying out the test, the same instructions are read to the students and the same indications are given so that the tests are consistent. The work carried out is part of the activities of the Advanced Research Laboratory in Artificial Intelligence and Human-Machine Interaction (LIAHM). LIAHM’s logo is shown in Figure 1. Figure 1: LIAHM’s logo. In this research paper, all theoretical background related are presented, as well as the system of equations used for the analysis, the results obtained and the discussion of such results. MALE AND FEMALE PERCEPTION OF BEAUTY Syntek (2020) explains in his video several issues related to male and female perception of beauty about men and women. Some initial considerations about beauty perception Symmetry in the perception of male and female beauty If you are a man, carefully observe the two faces of the twins in Figure 2a and indicate which one is the most beautiful. If you are a woman, observe the two faces of the twins in Figure 2b indicating which one is more attractive to you. Stop reading and take some time to do your choice. Which face is more beautiful? A or B? People always choose face A. When they are questioned as to why they chose that particular face, they answer that it is due to the eyebrows, lips or some other characteristic of the face. However, both faces in Figure 2a correspond to the same person as well as both faces in Figure 2b. Each pair of faces are not of twins. Why then the difference in the perception of beauty? These differences are due to the fact that each pair of faces were graphically manipulated in order to increase symmetry in face A and increase asymmetry in face B. Human beings have evolved to prefer symmetry in faces, which tends to allow them to choose healthier people. This is because of millions of years of human evolution. Apparently, the human brain has evolved to prefer symmetry and reject asymmetry. Services for Science and Education – United Kingdom 260 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. a. Two female twins b. Two male twins Figure 2: Symmetry in the perception of male and female beauty. Time and fashion in the perception of female beauty The perception of beauty not only depends on biological issues, but also cultural ones. For example, if people from different times are asked from a varying set of models which one they think is more beautiful, they are going to choose the one as close as possible to the clothing parameters of the time. Consider Figure 3, where three women in bikini are shown. The question is: which of the three models is more beautiful? This question is biased, because it depends on the time the person answering the question belongs to. Different people from different times will have different answers due to different cultural differences. URL: http://dx.doi.org/10.14738/assrj.99.13088 261 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Figure 3: Beauty perception across time for different cultural backgrounds A person in 1920 would have chosen model A, one in 1950 would have chosen model B and one in 1970 would have chosen model C. Body proportions in the perception of female beauty For the male brain, the ideal proportion between the waist and the heap is 70%, that is, the waist should have 70% of the size of the heap. This proportion is maintained from race to race and for different waist sizes, that is, consider more or less fatty women. From that comes the famous 90-60-90 proportion for the breast, waist and heap sizes. Notice that 60/90 ≈ 0.70. Apparently, this proportion indicates the male brain that the women being considered would be better at giving birth without complications (size of the heap) and possibly also holding babies easier. Services for Science and Education – United Kingdom 262 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. Figure 4: The ideal female proportion for the male brain The golden ration in the perception of female facial beauty Even more incredible that the previous issues related to the metrics of the human body, is the golden ratio, which has been considered a relation associated to beauty throughout human history. Apparently, the human brain seeks the golden ratio in the faces of women. Syntek (2020) and his team carried out experiments of their own with the faces of three models analyzing several attributes. As an example, consider two attributes: height and width of a face. It is assumed that having a face height divided by its width closer to the golden ratio is indicative of beauty. Voice in the perception of male and female beauty Syntek’s team (2020) also did a study in the perception of voice in groups of mean and women. They had the voice recorded of about three women and each individual in a group of men was asked to indicate which voice seemed more attractive. The same was done with men’s voices in a group of women. The results indicate that men prefer a higher pitch in women’s voices, whereas women prefer a lower pitch in men’s voices. This is probably due to hormonal changes affecting voice in women and men. A higher pitch in a female voice seems indicative of more sexual appealing due to the action of hormones in women’s voices. The same, but having a lower pitch in men’s voices seems to apply to women. Beauty approaches Historically, beauty has been the object of interest of philosophy, art, theology and science. However, it corresponds to aesthetics the study of the essence and perception of beauty (Serracanta, 2020), as well as its rules and methods to study it (Rodríguez et al., 2000). One of the fundamental problems of aesthetics is to define what constitutes beauty. However, defining the concept of beauty results complex and it leads to interminable discussions. It could be said that beauty definitions have been developed within two points of view: the objectivist point of URL: http://dx.doi.org/10.14738/assrj.99.13088 263 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 view dominating from the classic Greek period to the renaissance and the relativistic-subjective perspective extending from modern times to the present. The objectivist point of view Pythagorean philosophers are the main representatives of the objectivist point of view of beauty. This approach is based on the physical properties of the object: symmetry, balance and mathematical proportion of its parts. These properties or qualities, considered to be perfect, express the beauty of the object and generate emotions for everybody contemplating it. From this point of view, a face with perfect symmetry in its facial characteristics is considered to be beautiful. Realistic and formalistic aesthetic theories are based on the objectivistic point of view since they state that aesthetic judgements are universal. Relativistic-subjectivist approach to beauty From a subjectivist approach, the concept of beauty varies and it depends on the subject observing and judging. This approach manifests itself, above all, in the British culture, thanks to Locke’s conception concerning knowledge. To Hume, beauty “exists only in the mind of the one contemplating it and each mind perceives different beauty” (as quoted in De Bartolomeo & Magni, 2012, p. 2-3). The relativistic approach to beauty lies in the consideration that beautiful depends on the culture and historical period. Generally, for the subjective-relativistic approach, beauty is a subjective perception depending on multiple factors both individual and sociocultural. Thus, from this point of view it would be impossible to formulate an objective and universal concept of beauty, since it is denied that beauty is a quality of the object. Besides, Hume believes it is impossible to define a rule to apply to all countries and to all historical periods (De Bartolomeo & Magni, 2012). Biological basis of beauty perception The regions of the brain sensitive to the perception of beauty have been studied in human beings. The scientific evidence suggests that a network of brain regions including the accumbens nucleus, the fore cingular cortex, the medium prefrontal cortex and the orbitofrontal cortex are involved in the processing of attractive faces (Clouthier, Heatherton, Whalen, & Kelley, 2008). The findings of O’Doherty et at. (2003) using functional magnetic resonance imaging (fMRI) indicated that people activated the orbitofrontal medium cortex when perceiving an attractive face. It is a region involved in representing the stimulus-reward value. This indicates that when perceiving an attractive face, the rewards areas of the brain are activated. The same study reported that the response of this region of the brain improved with a smiling facial expression. On the other hand, Ishizu and Zeki (2011) analyzed the regions of the brain activated during auditive and visual perception. The results of their study indicated it is simply a cortical area located in the middle orbitofrontal cortex that is activated during the perception of beauty. Also, they observed that the activation force of that area to be proportional to the declared intensity of the experience of beauty. According to the sex of the person perceiving facial beauty, studies have revealed that regardless of the sexual orientation, both sexes perceive female or male facial attractiveness in a similar way (Kranz & Ishai, 2006). Services for Science and Education – United Kingdom 264 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. Concerning age, experimental studies have been carried out reporting that newborns prefer attractive faces (Slater, Quinn, Hayes, & Brown, 2000). Another study pointed out that young and middle age people evaluated the faces of all genders in the same way. However, the worst graded faces corresponded to old women (Foos & Clark, 2011). Facial beauty The different ideas about facial beauty have not only varied throughout history, but have frequently required the use of tricks for faces in order to achieve the ideal of beauty according to the respective epoch. Beauty parameter have been pointed out to be arbitrary cultural agreements. However, some authors such as Eco (2017, p. 14) consider that “it is possible that, beyond different beauty conceptions, there may be some unique rules for all peoples and all times”. It has been demonstrated that members of different ethnic groups share common attractiveness standards (Cunningham, Roberts, Barbee, Druen, & Wu, 1995). Other studies indicate that the opinions about facial beauty are consistent regardless of race, nationality or age (Fink & Neave, 2005). For Baudouin (2016) there are common criteria of beauty shared by all human beings. Although some aspects of judgment may reflect cultural conventions, the geometric characteristics of the human face originating beauty perceptions reflect universal beauty adaptations (Thornhill & Grammer, 1999). Some authors suggest that some facial preferences may be part of our biological rather than cultural inheritance (Rhodes, 2006). Some human facial characteristics considered to be attractive are mostly symmetry and the average shapes (Germine et al., 2015). In many species an asymmetry may be linked to a genetic anomaly or be shown in the individuals exposed to environmental problems (contamination, parasites or diseases). On this regard symmetry is an indicator of health and reproductive value of competitors (Baudouin, 2016). There is scientific evidence indicating that the preference for symmetrical faces may have an adaptive value (Scheib, Gangestad, & Thornhill, 1999, as quoted in Fink & Neave, 2005, p. 320). Concerning average characteristics, they refer to faces having common facial features or prototypes (Fink & Penton-Voak, 2016). The basic proportion of the human face is sexually dimorphic, that is, among men and women there are differences in their facial appearance depending on sexual hormones. Thus, for example, male features develop under the influence of testosterone, whereas female features develop under the influence of estrogens (Fink & Neave, 2005). Therefore, there are facial features typically male or typically female. Baudoin (2016) points out that for women in particular these features imply: big eyes, high and thin eyebrows, small nose and narrow jaw. On the other hand, for men, the characteristics are marked eyebrows closer to the eyes and smaller eyes. The same author considers people characterized as undefined are in the middle point of the male-female continuum. Another characteristic of facial beauty is related to the healthy look of skin. There is scientific evidence that the facial attractiveness of women is an indicator of hormonal health (Thornhill & Grammer, 1999). Dermatological studies show that dermatitis is associated to high levels of sexual hormones (Ghosh, Chaudhuri, Jain, & Aggarwal, 2014). Thus, for example, the polyquistic ovaric syndrome generates an overproduction of androgens clinically manifested as dermatitis URL: http://dx.doi.org/10.14738/assrj.99.13088 265 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 in women (Schiavone, Rietschel, Sgoutas, & Harris, 1983 as quoted in Fink & Neave, 2005, p. 319). Within this context the skin indicates, more or less reliably, the value of the female partner. In humans it is expected for men to be more sexually attracted towards women showing a skin free of lesions, eruptions, lunars, quits, tumors, acne and hirsutism (Fink & Penton-Voak, 2016). Concerning eye beauty, Baudouin and Tiberghien (2004) applied a metrical facial approach to the study of beauty and discovered that female facial attractiveness is higher when the face shows big eyes and thin eyebrows. Related to lips beauty, Etcoff (1999) holds that when women are young, their lips are red and burly. Such characteristics are indicators of youth and reproductive capacity. In puberty estrogens make the lips to thicken and redden whereas thin and flat lips indicate fragility and senility. THE GOLDEN RATIO (j) The golden ratio is a very intriguing number related to perceived beauty in nature. It is denoted using the Greek letter Phi (F) or the Greek letter phi (f), being these the uppercase and lowercase letters, respectively. We will simply use fi or phi (j). Definition of phi (j) Phi (j) is simply a ratio or proportion between two distances (Dobre, 2013). Figure 5 shows a #### must be equal to line AB #### divided by straight line and three points. Line #### AC divided by line AB ####, which equals j, as indicated in equation (1). line BC B A C Figure 5: Basic definition of j #### !" #### !$ = #### !$ #### $" =φ (1) Specifying phi (j) In order to specify the value of j it is necessary to use a bidimensional graphical context. Actually, j equals the longer distance of a specific rectangle divided by its shorter distance. Such rectangle is further divided in a square and another rectangle. This last rectangle has the same proportions of j as to the division of its longer distance by its shorter distance. And we can proceed further endlessly. This is illustrated in Figure 6. Services for Science and Education – United Kingdom 266 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. a a b Figure 6: Illustrating the Golden ratio in a series of rectangles We have that the horizontal distance of the larger rectangle (a+b) divided by the distance of the vertical side of the yellow square given by a, equals the vertical distance of the red rectangle given by a divided by its width given by b. This is mathematically expressed in equation (2). %&' % ='=φ (2) % Algebraically working out equation (2) yields equation (2a). % ' ' % +% =1+% ='=φ % Taking the right side of equation (2a) yields equation (3). % =φ ' Algebraically manipulating equation (3) yields equation (3a). ( ' = ) % (2a) (3) (3a) Notice that equation (3a), which is b/a, equals 1/j, so that substituting b/a for 1/j in equation (2a) yields equation (4). ( φ=1+) (4) Equation (4) is the fundamental equation for j. Calculating phi (j) But, what is the exact value of j? For this we consider equation (4) again. Both sides of such equation are multiplied by j, as indicated in equation (5). ( φ )φ = 1 + )* = φ* = φ + 1 (5) URL: http://dx.doi.org/10.14738/assrj.99.13088 267 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Equation (5) is equated to zero in equation (5a). φ* − φ − 1 = 0 (5a) Being j the unknown in equation (5a), it is possible to calculate the value of j using the quadratic formula, which is applied in equation (6). φ= (±√(&* = (±√. * (6) The negative part of the square root is discarded, so that j is defined in equation (7). Using the high precision scientific calculator of an iPhone yields the first digits of j as shown in equation (7). φ= (&√. * ≈ 1.618033988749895 (7) Figure 7: The infinite spiral of j The beauty of phi (j) The number j is related to a spiral infinite in nature, which can be expanded outwards and inwards. In order to appreciate it we have to work with the rectangle of Figure 6. Notice such rectangle is horizontal and that the square is on the left side. The vertical rectangle remaining on the right side can be divided into another rectangle in the superior side and then for the horizontal rectangle resulting in the inferior side, a square can be extracted on the right side, then from the vertical rectangle remaining we can extract another square on the inferior side, finally remaining another rectangle in the middle that is once again horizontal. And so on, it is possible to further make cuts in the rectangles, first on the left side, then the up side, then the right side, then on the down side to go back to the same sequence, left, up, right, down, and so Services for Science and Education – United Kingdom 268 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. on, which allows us to draw a spiral that reduces in size (and as extrapolation follow an inverse process for the spirals outside the page). This is illustrated in Figure 7. Figure 8: The eye of God What is important to notice is that the spiral touches the extreme points of the sequences of squares, which are highlighted in Figure 7. Observe Figure 8. The diagonals drawn on the rectangles are the golden ratio (gray line) marking what is known as the eye of God, which is where the spiral converges to in an infinitely small point. These lines cross the vertices of all golden ratio rectangles than can be drawn. The point where the lines intersect is known as the eye of God. These spirals are seen in nature and the ratios indicated by j are shown in all kinds of aesthetic situations, being those related to the beauty of the human body, to art, to nature, among a limitless number of applications. FACE PROPORTIONS TO CONSIDER AND THEIR MEASUREMENT The golden ratio is compared to a series of proportions of the face of a woman. A numerical score between zero and one is obtained which, when multiplied by ten, becomes a grade between zero and ten. Concepts of fuzzy logic are used for this purpose (Zadeh, 1965, 1997, 1999, 2002, 2005, 2008). Fuzzy logic is an area of study of artificial intelligence. A series of measurements of the face of each participant are made and then a grade between zero and ten is obtained in order to assess how close to the golden ratio such facial feature is. All grades are averaged using a weighted score in order to get a final grade for each participant. The literature was reviewed and considerations concerning facial beauty according to the golden ratio have been made (Packiriswamy, Kumar, & Rao, 2012; Alam, Mohd Noor, Basri, Yew, & Wen, 2015; Companioni Bachá, Torralbas Velázquez, & Sánchez Meza, 2010), as well as measurements of the skull considering as a reference the golden ratio (Bakirsi, Kafa, Coskun, URL: http://dx.doi.org/10.14738/assrj.99.13088 269 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Buyukuysal, & Barut, 2016a, 2016b; Suazo Galdames, Trujillo Hernández, Cantín López, & Zavando Matamala, 2008). Calculating a score Suppose we have a female face such as the one in Figure 9. This face is framed by a rectangle spanning it and two measurements are considered, the height of the rectangle, denoted by “y” as well as the width of the rectangle, denoted by “x”. y x Figure 9: A possible measurement of the face of a woman How close is the proportion y/x to the golden ratio j? If the ratio y/x were a perfect match to the golden ratio, then equation (8) would hold. / =φ (8) 0 Thus, in this case, the ratio y/x divided by j would be equal to one. Equation (9) shows the division between the ratio y/x and j. This division is denoted as “z”. It is possible that the ratio y/x to be approximately equal to j, in which case z ≈ 1, to be greater than j, in which case z > 1 or to be less than j, having z < 1. It is not possible for z to be equal to zero, since that would require a value for “y” to be zero, that is, not to have such measurement, which is not possible. //0 z= ) (9) Additionally, we would not expect z to be greater than two, since that would require y/x to be twice the value of j, which would be the measurement of a characteristic completely out of what is reasonable to expect. Consequently, we can calculate the value of f according to equation (10). 2 − z, 1 ≤ z ≤ 2 0≤z<1 f = 8 z, (10) 0, z>2 Services for Science and Education – United Kingdom 270 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. What is the fuzzy logic reasoning used in equation (10)? If z has a value of one, f would be equal to 2-1 = 1, that is, a perfect score of closeness to the golden ratio. If z is between one and two we would get a score between zero and one, depending on how far z is to the golden ratio. On the other hand, if z is between zero and one, the score is simply equal to z, since this yields a value between zero and one. Finally, if the proportion z = (y/x)/j is greater than two we have a case very far from the golden ratio, so that the score associated to such measurement would be equal to zero. The numerical grade of the proportion for the face, denoted as “c”, equals ten time the value of f, as indicated in equation (11). c = 10f (11) Weighting several scores corresponding to several measurements Suppose there are different ratios for the face of some woman. In this case, there are m values of y, that is, yk, k = 1, 2, …, m, and m values for xk, k = 1, 2, …, m, where k is the kth measurement. Notice that in all measurements the value of yk must be associated to the measurement expected to be the highest and the value of xk to that expected to be lowest. In this way, we get scores zk, k = 1, 2, …, m, according to equation (12). / /0 z2 = !) ! , k = 1, 2, …, m (12) The fuzzy logic grade for zk is calculated according to equation (13), which follows the same logic as equation (10). 2 − z2 , 1 ≤ z2 ≤ 2 0 ≤ z2 < 1, k = 1, 2, …, m f2 = 8 z2 , (13) 0, z2 > 2 The zero to ten grade, denoted by ck, for the fuzzy logic score fk, equals ten times the value of fk, as indicated in equation (14). ck = 10fk, k = 1, 2, …, m (14) Finally, and assuming the weight assigned to the kth measurement is given by wk, the final score is given by equation (15). Equation (16) must hold, that is, the sum of the weights must be 100% of the final score or grade. c = ∑3 (15) 24( w2 c2 ∑3 24( w2 = 1 (16) Measurements to do What ratios of the face of a woman should be considered? We seek those proportions that tend to approach the golden ratio of a beautiful face. The first measurement to do is the height of the face (y1) with respect to the width of the face (x1), which has already been indicated in Figure 9. The second measurement to do is the distance between the forehead and the tip of the nose URL: http://dx.doi.org/10.14738/assrj.99.13088 271 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 (y2) and the distance between the tip of the nose and the chin (x2), as indicated in Figure 10. The third measurement is the distance between the eyes (y3) and the width of the mouth (x3) as indicated in Figure 11. Finally, the fourth measurement to do is the distance between the top of the nose to the mouth (y4) and the distance between the mouth and the chin (x4), illustrated in Figure 12. Notice that, for having a total of four measurements for each woman, m = 4. y2 x2 Figure 10: Second measurement to do per woman y3 x3 Figure 11: Third measurement to do per woman. Services for Science and Education – United Kingdom 272 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. y4 x4 Figure 12: Fourth measurement to do per woman We are going to work with three women. The images of the three women are shown in Figure 13. Figure 13a shows the first such woman, Figure 13b shows the second such woman and Figure 13c shows the third such woman. a. Woman 1. b. Woman 2. c. Woman 3. Figure 13: The three women to consider. Based on these images the measurements to calculate the adjustment to the golden ratio for each woman are taken. We assume the four measurements have the same weight, that is, w1 = w2 = w3 = w4 = 0.25. Table 1 summarizes all relevant information. URL: http://dx.doi.org/10.14738/assrj.99.13088 273 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Table 1: Calculating the adjustment to the golden ratio for the three women. Concept Woman 1 Woman 2 Woman 3 y1 4.0 4.5 4.3 x1 3.3 3.5 3.1 y1/x1 1.2121 1.2857 1.3871 z1 0.7491 0.7946 0.8573 f1 0.7491 0.7946 0.8573 c1 7.5 7.9 8.6 y2 2.5 2.8 2.8 x2 1.7 1.6 1.5 y2/x2 1.4706 1.7500 1.8667 z2 0.9089 1.0816 1.1537 f2 0.9089 0.9184 0.8463 c2 9.1 9.2 8.5 y3 1.7 1.6 1.7 x3 1.1 1.2 1.1 y3/x3 1.5455 1.3333 1.5455 z3 0.9551 0.8240 0.9551 f3 0.9551 0.8240 0.9551 c3 9.6 8.2 9.6 y4 1.6 1.7 1.7 x4 1.0 1.0 0.8 y4/x4 1.6000 1.7000 2.1250 z4 0.9889 1.0507 1.3133 f4 0.9889 0.9493 0.6867 c4 C j= 9.9 9.0 1.618034 9.5 8.7 6.9 8.4 The best grade is for the first woman (C = 9.0), the second-best grade is for the second woman (C = 8.7) and the worse grade is for the third woman (C = 8.4). With these grades or scores, it is possible to compare the order in the degree of beauty of the three women according to how closely they match the golden ratio with respect to what two groups of test subjects consider how beautiful each woman is (the first group of test subjects for men, the second one for women). Certainly, in order to measure how close to the score obtained in Table 1 for each woman the test subjects consider these women to be, requires a quantitative and statistical way of measuring the degree of closeness between what the golden ratio suggests and what is obtained in the experiments carried out. Services for Science and Education – United Kingdom 274 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. EXPERIMENTAL DESIGN In order to prove (or deny) the theory that a female face is beautiful if it adjusts, statistically speaking, to the parameters indicated by the golden ratio, a series of statistical tools and considerations are required. First, there is a general score calculated from the m measurements carried out for each of the three faces of the female models (women) selected. Let C1 be the highest score obtained by one of the three contestants, C2 the score in between and C3 the lowest score obtained. A series of experiments are carried out using a graphic interface specifically developed in order to do the statistical measurements. In this case, we consider two groups of individuals separately: men and women. Let n be the total number of experimental data obtained by having men (or women) judging the facial beauty of the contestants. We have two sets of experimental data of n pairs of measurements each. Notice that the number of n men doing the subjective evaluations does not necessarily have to be the same number of n women doing the same subjective evaluations. However, for simplicity, we speak of n measurements in each series of experiments. We consider two pieces of information: the number of seconds the person took to reach its subjective evaluation of the beauty of the female models (y), as well as the score obtained in such subjective evaluation with respect to how close was the evaluation compared to the suggested order given by the pattern indicated by the adjustment to the golden ratio of the facial beauty of the three female models (x). The time calculation (in seconds) for the amount of time each experiment lasts for each test subject is trivial to obtain. However, the calculation of the score of adjustment to the facial beauty as indicated by the golden ratio is not. Calculating the experimental adjustment score to the pattern suggested by the golden ratio criteria According to the score obtained considering the adjustment of each of the three faces of the female models to the golden ratio, it is possible to sort the three faces, from left to right, going from highest to lowest score, respectively. Thus, the female face with the highest score is denoted as 1, the one with the middle score is 2 and the one with the worse score is 3. Then, the “optimal” sequence according to the subjective grade of the faces of the three female models is 1-2-3. The test subject is presented with the three faces not sorted from left to right as best to worst score. How many possible sortings can we have? We have to find the number of permutations of n faces taken by r per r. Such number of permutations (Walpole & Myers, 1989), denoted as nPr, is indicated according to equation (17). Notice that this n has nothing to do with the n pair of experimental results (one for men and the other for women). 5! (17) n Pr = (589)! In this case, we have 3 permutations taken by 3 per 3, that is, n = 3 and r = 3, so that 3P3 = 3!/(33)! = 3!/0! = 6/1 = 6. Nevertheless, which ones are these six permutations? The experimental subject can choose the faces in the following ways: 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2 and 3-2-1. Let T1 be the priority assigned to the face considered by the test subject to be the most beautiful (where T1 = {1, 2, 3}), T2 the priority assigned by the test subject to be the medium best-looking face (where T2 = {1, 2, 3}), and T3 the priority assigned by the test subject to be the worst looking face (where T3 = {1, 2, 3}). The formula used to calculate the score (Si, i = 1, 2, …, n) for the URL: http://dx.doi.org/10.14738/assrj.99.13088 275 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 selection of each test subject with respect to what is suggested by the fitness to the golden ratio is indicated in equation (18), where n is the total number of measurements carried out (in one case for men test subjects and in the other for women test subjects). Si = (T1-1)2+(T2-2)2+(T3-3)2, i = 1, 2, …, n (18) Does the formula for equation (18) make sense? Let us see. If test subject i grades the models as 1-2-3 (that is, exactly according to what the golden ratio suggests), it would have a score of Si = (1-1)2+(2-2)2+(3-3)2 = 0. This is the best score than can be obtained. If, on the other hand, the test subject i grades the models as 3-2-1 (that is, in a completely reversed order, because the worst looking model goes first and the best-looking model goes last), the score obtained would be Si = (3-1)2+(2-2)2+(1-3)2 = 8. A modestly bad score would be, for example, 3-1-2, that is, Si = (3-1)2+(1-2)2+(2-3)2 = 6. Consequently, the best score possible is zero and the worst score possible is eight, having as intermediate scores two and six. In order to get a numerical score between zero and one, Pi, we use equation (19). (<8= ) P; = < " , i = 1, 2, …, n (19) This score (Pi) is the one used as first variable (xi, i = 1, 2, …, n). We can have values for xi, i = 1, 2, …, n, of (8-0)/8 = 1, (8-2)/8 = 6/8 = 3/4 = 0.75, (8-6)/8 = 2/8 = 1/4 = 0.25 and finally (8-8)/8 = 0. The second variable (yi, i = 1, 2, …, n) is the number of seconds it took test subject i to sort the models. Therefore, for each set of n data (one for male experimental subjects and the other for female experimental subjects, where the values for n are not necessarily the same), we have two sets of results, (xi,yi), i = 1, 2, …, n. y (x4,y4) (x1,y1) (x2,y2) (x3,y3) (x5,y5) x Figure 14: Perfect linear adjustment with positive slope (r = 1) and n = 5 data points. Linear adjustment and the linear correlation coefficient Generally speaking, if the we use scatterplots for two variables (x,y), we try to fit the results using a straight line as well as to evaluate how good is such adjustment using the linear correlation coefficient (r). If r = 1, the data fit perfectly a straight line with a positive slope (see Figure 14), if r = -1, it means the data fit perfectly a negative slope (see Figure 15) and if r = 0, Services for Science and Education – United Kingdom 276 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. there is no adjustment, meaning all the data is completely scattered for the entire (x,y) scatterplot. y (x3,y3) (x1,y1) (x4,y4) (x5,y5) (x2,y2) x Figure 15: Perfect linear adjustment with negative slope (r = -1) and n = 5 data points. Nevertheless, what happens in realistic cases where the data does not perfectly fit a straight line? Such situation is illustrated in Figure 16. In this figure, we try for the straight line to fit in such way so as to minimize the errors (distance) between the points and the straight line adjusting them. y (x4,y4) (x2,y2) (x5,y5) (x1,y1) (x3,y3) x Figure 16: Optimal adjustment for a straight line having five (n = 5) realistic experimental results. How can we get the parameters of such straight line for optimal adjustment? The general formula for all straight lines is given in equation (20), where a is the point at which the line cuts the vertical axis and b is the slope of such straight line. y = a + bx (20) Following some mathematical work that needs no demonstration here, we get the value of a and b as indicated in equations (21) and (22), respectively. a= ∑ 0# ∑ /8∑ 0 ∑ 0/ 5 ∑ 0# 8(∑ 0)# URL: http://dx.doi.org/10.14738/assrj.99.13088 (21) 277 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 b= ? ∑ 0/8∑ 0 ∑ / 5 ∑ 0# 8(∑ 0)# (22) How well these two parameters (a and b) fit the experimental data? Walpole and Myers (1989) indicate that the straight line described by equation (20) fits the parameters from equations (21) and (22) having a linear correlation coefficient, r, where -1≤r≤1, given by equation (23). r= ∑$ "%& @" A" # $ # B∑$ "%& @" ∑"%& A" (23) We have for the case of equation (23) that Xi and Yi are given by equations (24) and (25), respectively, having x# and y# given by equations (26) and (27), respectively. Notice that x# and y# are the mean (average) values of the data. (24) Xi = xi-x#, i = 1, 2, …, n Yi = yi-y#, i = 1, 2, …, n x# = y# = (25) ∑$ "%& 0" 5 (26) ∑$ "%& /" 5 (27) Also, we have that the variances of the variables x and y, sx2 and sy2, respectively, are given by equations (28) and (29), respectively. Notice that if we consider the population mean for both variables (µx and µy) to be properly represented by the means of equations (26) and (27), and if the data is normally distributed, 95% of all the data would be between µx-2sx and µx+2sx for x and between µy-2sy and µy+2sy for y. Realize that the set of data “y” is given by the time it takes the experimental subject to solve the problem, whereas the set of data “x” is given by the score (Pi) each individual obtains in his/her subjective evaluation of the facial beauty of the three female models. σ*0 = σ*/ = ∑$ # )# "%&(0" 80 58( (28) ∑$ C )# "%&(/" 8/ 58( (29) Designing the user interface (software) in order to carry out the experiments A specially designed software was created in order to carry out the experiments with men and women. In Figure 17 we show the interface for experiments with men (it says “Prueba para Hombres” meaning “Test for Men”). In the case of the interface for women, it is the same, but it says “Prueba para Mujeres”, meaning “Test for Women”. Services for Science and Education – United Kingdom 278 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. Figure 17: Experimental interface. RESULTS We had a total of 88 men and 109 women doing the golden ratio experiment. In both cases, the independent variable (x) was the grade or score obtained according to equation (19), which is a number going from zero to one, but can only take specific values of 0, 0.25, 0.75 and 1. The dependent variable (y) was the time each individual took to do the test, measured in seconds. A Pearson correlation between the two variables (x,y) was carried out. Also, a linear regression was conducted. Coefficients a and b for the linear regression (linear correlation) are given by equations (21) and (22), respectively. The Pearson correlation coefficient is given by equation (23), where Xi, Yi, x# and y# are given according to equations (24), (25), (26) and (27), respectively. Table 2 summarizes the results obtained for men. Table 3 indicates the results obtained for women. URL: http://dx.doi.org/10.14738/assrj.99.13088 279 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Table 2: Results of the experiments carried out for men. n= Sx = Sy = 88 29.25 2324 Sx2 = Sxy = x" = y" = 19.3125 818 0.33 26.4091 SX2 = 9.59 SY2 = SXY = a= b= r= 15387.27 45.53 24.831 4.748 0.1185 R2 = r2 = 0.0141 Tabla 3: Results of the experiments carried out for women. n= Sx = Sy = 109 55.25 2955 Sx2 = Sxy = x" = y" = 43.8125 1646.75 0.51 27.1101 SX2 = 15.81 SY2 = SXY = a= b= r= 26672.68 148.92 22.335 9.4208 0.2293 R2 = r2 = 0.0526 In order to deepen the analysis and to clarify what is happening, descriptive statistics are included. Table 4 shows the calculation for minimum and maximum, mean or average, T.D. or typical deviation (also known as standard deviation, which equals the square root of the variance), mode and median. Services for Science and Education – United Kingdom 280 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. Variable x Men y Men x Women y Women Minimum 0 2 0 1 Table 4: Descriptive statistics Maximum Mean T.D. 1 0.3324 0.3320 77 26.4091 13.2991 1 0.5069 0.3826 97 27.1101 15.7153 Mode 0 17 0.25 16 Median 0.25 24 0.75 23 The descriptive statistics for the values of x are analyzed considering the number of times each one of the possible values (0, 0.25, 0.75 and 1) occur, which is shown in Table 5 and Table 6 for men and women, respectively. Table 5: Score distribution for men. x Men Totals Percentage 0 31 35.23% 0.25 29 32.95% 0.75 24 27.27% 1 4 4.55% 0.3324 88 100.00% Table 6: Score distribution for women. x Women Totals Percentage 0 23 21.10% 0.25 31 28.44% 0.75 30 27.52% 1 25 22.94% 0.5069 109 100.00% DISCUSSION AND CONCLUSION The data for Table 2 indicates than there is a positive relationship between the score (x) and the time used for the test (y). In the case of men, such relationship is given by equation (30), whereas in the case of women the relationship is given by equation (31). y = 4.748x + 24.831 (30) y = 9.4208x + 22.335 (31) However, actually, the score should be the dependent variable (it is the independent variable or x) and the time used should be the independent variable (it is the dependent variable or y). Thus, solving for x in both equation (30) and equation (31) yields equations (32) and (33) for the cases of men and women, respectively. x = 0.2106y – 5.2298 (32) x = 0.1061y – 2.3708 URL: http://dx.doi.org/10.14738/assrj.99.13088 (33) 281 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 Both in the case of equations (30) and (31) as in the case of equations (32) and (33) the Pearson correlation coefficient (r) is the same. In this case, there is a positive correlation for both variables of r = 0.1185 for men and r = 0.2293 for women (the correlation does not change regardless of what variables are considered independent or dependent). The slope for both equations is always positive. This simply means that the more time it takes to reach a decision, the better the decision is. First, in Table 4, there is a summary of basic descriptive statistics. In this case, we have the values for the minimum, maximum, mean, typical deviation or standard deviation, mode and median. Notice that in the case of the score calculated (which is different from the golden ration score for each model shown in Table 2), there are discrete instances for x, that is, x can be equal to 0, 0.25, 0.75 or 1. It is not a surprise to see that the mode for the values of x in Table 4 corresponds to the most common value (0 in the case of men, 0.25 in the case of women). The median is simply the value in the middle or the sorted list of the values obtained. In the case of men such value is 0.25, whereas in the case of women such value is 0.75. The mean for men is 0.3324 whereas for women it is 0.5069. It seems that in the case of men the selections were contrary to what the golden ratio suggests. Apparently, men got carried away by stereotypes in their decision. Notice that according to our golden ratio measurements, the most beautiful model is number 1, followed by model number 2 and finally model number 3, as shown in Figure 13. However, in the interface shown in Figure 17 the most beautiful woman is shown last, the least beautiful model is in the middle, whereas the model in between with respect to its fit to the golden ration appears first in the interface. Consequently, since the model that appears first in the interface is blond and the model that appears last in the interface (being the most beautiful according to the fit to the golden ratio measurements) has big ears, it seems men acted impulsively without concentrating on the face but rather following stereotypes and following the impulse to click on the finalize button immediately without giving enough consideration to the faces instead of the overall look of the model. All test subjects were given specific instructions to focus on the beauty of the face and not to follow stereotypes. On the other hand, women did a much better job and considered the instructions carefully. All four possible scores in the case of women are approximately equally likely and, as a consequence, the mean approximately equals 0.50. It seems men perceive beauty in a more visual way, whereas women perceive beauty more verbally. Men seem to consider beauty judging by looks and visual stereotypes, whereas women consider beauty based on what they are told is beautiful. In order to get a score of 1 the minimum effort requires doing two face swapes using the interface, which requires considerable effort and time dedicated to the experiment. Thus, the time taken to do the experiment is relevant and it makes sense to see that the greater the amount of time taken to do the experiment, the better the decision reached. As such, what about the analysis for the time to respond? Although the response time is a discrete variable (seconds), it has considerable variation. In the case of men, according to Table 4, it varies between 2 seconds and 77 seconds, whereas in the case of women it varies between 1 second and 97 seconds. Notice that women on average took 27.1101 seconds, whereas men took less time, 26.4091 seconds. Thus, women tend to take more time to decide which should be the order of the models and tend to reach a better result when doing the experiment, which coincides with the theory of the golden ratio. Services for Science and Education – United Kingdom 282 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. It is concluded that women tend to make a better decision than men, possibly due precisely to the instructions given consisting in only considering the interior of the face, without considering ears or hair color, and to avoid prejudice such as “if it is blond is more beautiful”. For men, it seems the first impression, leading to avoid making changes in the order of the faces given by the interface and tending to use little time to do the experiment is what prevails. Apparently, men tend to decide female facial beauty more based on first impression visual cues, tending to prejudice, while women tend to decide female facial beauty in a more verbal fashion, trying to go deeper in their decisions. The parameters of beauty suggested by the golden ratio are more compatible to the perception of female facial beauty in the case of women than in men. The results should also be considered from another point of view. There is a total of 6 possible outcomes for the score obtained, both in men as in women, as indicated in equation (17). Of these six possibilities, only one, the 1/6 ≈ 0.1667 ≈ 16.67% would be the case for a perfect score of one by ordering the faces correctly as 1-2-3 (first model 1 as the most beautiful, followed by model 2 and then 3). That is, a perfect score of one corresponds to the order indicated if the golden ratio is a perfect benchmark for the facial beauty of the models. All other scores do not follow the score obtained if the golden ratio is an accurate indicator of facial beauty. All other possible scores yield a value less than one. Thus, the perfect score would occur only in one out of six possible outcomes, that is, the perfect score would be 1/6 ≈ 0.1667 ≈ 16.67% of the cases. In all other cases the golden ratio would not be followed. Consequently, having an average score above 0.1667 is already an indicator that the golden ratio does follow an important role, since in all the remaining 5 possibilities the golden ratio is not considered to be the indicator with the highest score. As a consequence, the fact that men have obtained an average score of 0.33 and women an average of 0.51 is already an indicator that the golden ratio does follow an important role in determining female facial beauty. To see things from this other point of view, consider throwing a dice. A dice can have as outcomes: 1, 2, 3, 4, 5 and 6. A truly random behavior would be to get probabilities for each of these possible outcomes of 1/6 ≈ 0.1667. Any other result above this probability for any of the outcomes would indicate that the sides of the dice are loaded, that is, that certain possible result is more relevant. As such, an outcome above 0.1667 for the value obtained in the experiments carried out would be indicative that the golden ratio is affecting the outcomes. From this alternative point of view, we could consider that the golden ratio plays a fundamental role in the perception of beauty of the test subjects. Thus, we would conclude that the golden ratio is a relevant factor indicative of female facial beauty both in men as in women. More specifically, it indicates that women are more sensitive than men to female beauty and that the golden ratio is important in such perception. Why are women more sensitive than men to the parameters of the golden ratio? Maybe it is because women are more responsive to issues related to facial beauty. It could also be argued that women are evolutionary more advanced than men. Nevertheless, all of the above would be highly speculative. What can indeed be stated is that the golden ratio substantially affects the perception of female facial beauty both in men as in women, but more for women than men. As to the debate about whether beauty is objective and has to do with issues related to proportions, which is the idea of this paper, or if it is completely relative and depends entirely on environmental issues, it is better to leave that to the judgment of the reader. Keep in mind, URL: http://dx.doi.org/10.14738/assrj.99.13088 283 Advances in Social Sciences Research Journal (ASSRJ) Vol. 9, Issue 9, September-2022 however, that according to the literature review of section 2, an innate biological component in the perception of beauty and even to regions of the brain doing such perception cannot be denied. References Alam, M. K., Mohd Noor, N. F., Basri, R., Yew, T. F., & Wen, T. H. (2015). Multiracial Facial Golden Ratio and Evaluation of Facial Appearance. PLOS One, 10: 1-22. Bakirci, S., I., Kafa, M., Coskun, I., Buyukuysal, M. C., & Barut, C. (2016a). A Comparison of Anatomical Measurements of the Infraorbital Foramen of Skulls of the Modern and Late Byzantine Periods and the Golden Ratio. International Journal of Morphology, 34: 788-795. Bakirci, S., I., Kafa, M., Coskun, I., Buyukuysal, M. C., & Barut, C. (2016b). A Comparison of the Relationship between the Golden Ratio and Anatomical Characteristics of the Supraorbital Foramen in Bare Skulls Belonging to the Byzantine Era and Modern Era. International Journal of Morphology, 34: 671-678. Baudouin, J. Y. (2016). “Que révèle le visage?”. Retrieved from: https://www.fichierpdf.fr/2016/05/05/que_revele_le_visage_jean_yves_baudoin/que_revele_le_visage_jean_yves_baudoin.pdf. Baudouin, J. Y. & Tiberghien, G. (2004). Symmetry, averageness, and feature size in the facial attractiveness of women. Acta Psychol. (Amst.), 117: 313-332. Clouthier, J., Heatherton, T. F., Whalen, P. J., & Kelley, W. M. (2008). Are attractive people rewarding? Sex differences in the neural substrates of facial attractiveness. J. Cog Neurosci, 20: 1-18. Companioni Bachá, A. E., Torralbas Velazquez, A., & Sánchez Mesa, C. (2010). Relación entre la proporción áurea y el índice facial en estudiantes de Estomatología de La Habana. Revista Cubana de Estomatología, 47: 50-61. Cunningham, M. R., Roberts, A. R., Barbee, A. P., Druen, P. B., & Wu, C. H. (1995). Their ideas of beauty are, on the whole, the same as ours: Consistency and variability in the cross-cultural perception of female physical attractiveness. Journal of Personality and Social Psychology, 68: 261-279. De Bartolomeo, M. & Magni, V. (2012). La fondazione dell´estetica. Istituto Italiano Edizioni Atlas. Dobre, D. (2013). Mathematical Properties of the Golden Ratio – a Fascinating number. Journal of Industrial Design and Engineering Graphics, 8: 11-16. Eco, U. (2017). Historia de la belleza. Debolsillo. Etcoff, N. (1999). Survival of the prettiest. Doubleday. Fink, B. & Penton-Voak, I. S. (2016). Evolutionary psychology of facial attractiveness, Curr. Dir. Psychol. Sci, 11: 154-158. Fink, B. & Neave, N. (2005). The biology of facial beauty. International Journal of Society of Cosmetic Science, 27: 317-325. Foos, P. W. & Clark, M. C. (2011). Adult age and gender differences in perceptions of facial attractiveness: Beauty is in the eye of the older beholder. The Journal of Genetic Psychology, 172: 162-175. Germine, L., Russell, R. , Bronstad, M. , Blokland, G. A. M., Smoller, J. W., Kwok, H., Anthony, S. E., Nakayama, K., Rhodes, G., & Wilmer, J. B. (2015). Individual aesthetic preferences for faces are shaped mostly by environments, not genes. Current Biology, 25: 2684-2689. Ghosh, S., Chaudhuri, S., Jain, V. K., & Aggarwal, K. (2014). Profiling and hormonal therapy for acne in women. Indian Journal of Dermatology, 59: 107-115. Ishizu, T. & Zeki, S. (2011). Toward a brain–based theory of beauty. PLoS ONE, 6: e21852. Kranz, F. & Ishai, A. (2006). Face perception is modulated by sexual preference. Curr Biol. 16: 63-68. O´Doherty, J., Winston, J., Critchley, H., Perret, D., Burt, D. M., & Dolan, R. J. (2003). Beauty in a smile: the role of medial orbitofrontal cortex in facial attractiveness. Neuropsychologia, 41: 147-155. Services for Science and Education – United Kingdom 284 Copertari, L. F., & Reyna, G. V. (2022). Beauty and the Golden Ratio. Advances in Social Sciences Research Journal, 9(9). 259-285. Packiriswamy, V., Kumar, P., & Rao, M. (2012). Identification of Facial Shape by Applying Golden Ratio to the Facial Measurements: An Interratial Study in Malaysian Population. North American Journal of Medical Sciences, 4: 624-629. Rhodes, G. (2006). The evolutionary psychology of facial beauty. Annu. Rev. Psychol, 57: 199-226. Rodríguez, M., Rodríguez, M. E., Barbería, E., Durán, J., Muñoz, M., & Vera, V. (2000). Evolución histórica de los conceptos de belleza facial. Ortodoncia Clínica, 3: 156-163. Serracanta, F. (2020). “Concepto de belleza – Historia de la sinfonía”. Retrieved from: https://www.historiadelasinfonia.es/monografias/las_sinfonias_de_khrennikov/concepto_de_belleza/. Slater, A., Quinn, P. C., Hayes, R. & Brown, E. (2000). The role of facial orientation in newborn infants´ preference for attractive faces. Developmental Science, 3: 181-185. Suazo Galdames, I., Trujillo Hernández, E. G., Cantín López, M., & Zavando Matamala, D. (2008). Determinación de Proporciones Áurea Cráneofaciales para la Reconstrucción con Fines de Identificación Médicolegal. International Journal of Morphology, 26: 331-335. Syntek, A. (2020). “ASOMBROSA-MENTE. Las Reglas de la Atracción”. Retrieved from: https://www.youtube.com/watch?v=zMm6QXEgpxU. Thornhill, R. & Grammer, K. (1999). The body and the face of woman: One ornament that signals quality? Evolution and Human Behavior, 20: 105-120. Walpole, R. E. & Myers, R. H. (1989). Probabilidad y Estadística para Ingenieros. McGraw-Hill. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8: 338-353. Zadeh, L. A. (1997). Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems, 90: 111-127. Zadeh, L. A. (1999). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 100: 9-34. Zadeh, L. A. (2002). Toward a perception based theory of probabilistic reasoning with imprecise probabilities. Journal of Statistical Planning and Inference, 105: 233-264. Zadeh, L. A. (2005). Toward a generalized theory of uncertainty (GTU) – an outline. Information Sciences, 172: 140. Zadeh, L. A. (2008). Is there a need for fuzzy logic? Information Sciences, 178: 2751-2779. URL: http://dx.doi.org/10.14738/assrj.99.13088 285