## The relation between golden ratio and beauty of human faces

Mathematics Internal Assessment

The ancient Greek philosopher once said, “Beauty lies in the eyes of the beholder”. For eons, beauty has been considered relative; something that cannot be measured due to the variance in its perception from individual to individual. But recently, as I was scrolling through my social media feed, I chanced upon a list of ‘Most Beautiful Women of 2017’, published by a popular webpage. This got me thinking about how there were certain individuals who are widely accepted as ‘beautiful’ in spite of the ambiguity of the very idea of ‘beauty’. I surfed the web, and found websites outlining how the golden ratio is rooted in the dimensions of the faces of many individuals who are considered attractive. I was intrigued, and decided to conduct an investigation on the correlation between the Golden Ratio and the beauty of a face.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities . The ratio is an irrational number, 1.61803….., but is often rounded off to 1.62 for convenience. The Golden ratio has been studied for thousands of years by mathematicians including Euclid, Fibonacci and Kepler. It is prevalent in nature, in the arrangement of flower petals, the seed pods of pinecones, as well as the dimensions of shells. The Golden Ratio is also present in many architectural structures such as the Egyptian pyramids, the Athena Parthenon and many more.

I selected the front profiles of three women from the internet, and calculated the following dimensions of their face.

a = Top-of-head (1) to chin (2)
b = Top-of-head (1) to pupil (3)
c = Pupil (3) to nosetip (4)
d = Pupil (3) to lip (5)
e = Width of nose (6 to 7)
f = Outside distance between eyes (8 to 9)
g = Width of head (10 to 11)
h = Hairline (12) to pupil (3)
i = Nosetip (4) to chin (2)
j = Lips (5) to chin (2)
k = Length of lips (13 to 14)
l = Nosetip (4) to lips (5)

I then calculated the following ratios.

a/g= Top-of-head (1) to chin (2)/ Width of head (10 to 11)
b/d= Top-of-head (1) to pupil (3)/ Pupil (3) to lip (5)
i/j= Nosetip (4) to chin (2)/ Lips (5) to chin (2)
i/c= Nosetip (4) to chin (2)/ Pupil (3) to nosetip (4)
e/l= Width of nose (6 to 7)/ Nosetip (4) to lips (5)
f/h= Outside distance between eyes (8 to 9)/ Hairline (12) to pupil (3)
k/e= Length of lips (13 to 14)/ Width of nose (6 to 7)

After calculating these ratios, I calculated the deviation of each of these ratios from the rounded off value of the Golden ratio, that is, 1.62. I then found the absolute sum of all the deviations.

I made a survey form in which 50 participants were asked to rate each of the three faces I made calculations on, on a scale of 1 to 5, 1 being the least attractive and 5 being the most attractive. I calculated the average score for each face.

I then found the correlation coefficient in order to establish a relationship between the average score for each face, and the deviation from the golden ratio, using the formula

It was found that there was a highly negative correlation, and that a higher score corresponded to lesser deviation from the Golden Ratio.

To check the reliability of the result, I used the Chi-Square test, using the formula

, where O= frequencies observed, and e= frequencies expected.

I then drew a probability distribution function to graph the results.

The three faces I used for my exploration are-
F1 F2 F3

Calculations

F1

Ratios of dimensions of face

1) a/g= 1.818
2) b/d= 1.7
3) i/j= 1.5
4) i/c= 1.23
5) e/l= 1.222
6) f/h= 1.28
7) k/e= 0.63
Deviation of ratios from Golden ratio

1.818-1.62=0.198
1.7-1.62=0.08
1.5-1.62=I-0.12I=0.12
1.23-1.62=I-0.39I=0.39
1.222-1.62=I-0.398I=0.398
1.28-1.62=I-0.34I=0.34
0.63-1.62=I-0.99I=0.99

Absolute sum of deviations= 0.198+0.08+0.12+0.39+0.398+0.34+0.99= 2.906

F2

Ratios of dimensions of face

a/g= 1.61
b/d= 1.61
i/j= 1.5
i/c= 1.22
e/l= 1.4
f/h= 1.147
k/e= 0.57

Deviation of ratios from Golden ratio

1.61-1.62= I-0.01I=0.01
1.61-1.62= I-0.01I=0.01
1.5-1.62=I-0.12I=0.12
1.22-1.62=I-0.4I=0.4
1.4-1.62=I-0.22I=0.22
1.147-1.62=I-0.473I=0.473
0.57-1.62=I-1.05I=1.05

Absolute sum of deviations= 0.01+0.01+0.12+0.4+0.22+0.473+1.05= 2.283

F3

Ratios of dimensions of face

a/g= 1.813
b/d= 1.87
i/j= 1.46
i/c= 1.05
e/l= 1.57
f/h= 1.03
k/e= 0.545

Deviation of ratios from Golden ratio

1.813-1.62= 0.193
1.87-1.62= 0.25
1.46-1.62=I-0.16I=0.16
1.05-1.62=I-0.57I=0.57
1.57-1.62=I-0.05I=0.05
1.03-1.62=I-0.59I=0.59
0.545-1.62=I-1.075I=1.075

Absolute sum of deviations= 0.193+0.25+0.16+0.57+0.05+0.59+1.075= 2.888

I created a frequency distribution table of the results from the survey.

Faces Frequency of the scores
1 2 3 4 5
F1 8 24 18 0 0
F2 0 0 10 23 17
F3 2 23 25 0 0

Average score for F1= ((8X1)+(24X2)+(18×3))/50= 2.2
Average score for F2= ((10X2)+(23X4)+(17X5))/50= 4.14
Average score for F3= ((2X1)+(23X2)+(25X3))/50= 2.46

I then found the Pearson correlation coefficient, to establish the relationship between the average score of the face and the deviation from the golden ratio.

Deviation=x
Average score=y
Total no. of faces=n

n=3
x= 2.906+2.283+2.888=8.077
y=2.2+4.14+2.46=8.8
xy1=2.906X2.2=6.3932
xy2=2.283X4.14=9.45162
xy3=2.888X2.46=7.10448
xy=xy1+xy2+xy3=6.3932+9.45162+7.10448=22.9493
〖x1〗^2=〖2.906〗^2=8.444836
〖x2〗^2=〖2.283〗^2=5.212089
〖x3〗^2=〖2.888〗^2=8.340544
x^2=〖x1〗^2+〖x2〗^2+〖x3〗^2=8.444836+5.212089+8.340544=21.997469
〖y1〗^2=〖2.2〗^2=4.84
〖y2〗^2=〖4.14〗^2=17.1396
〖y3〗^2=〖2.46〗^2=6.0516
 y^2=〖y1〗^2+〖y2〗^2+〖y3〗^2=4.84+17.1396+6.0516=28.0312

Substituting the values in the equation,

r=(3(22.9493)-(8.077)(8.8))/√([(3X21.997469)-〖8.077〗^2 ][(3X28.0312)-〖8.8〗^2])
=(68.8479-71.0776)/√([65.992407-65.237929][84.0936-77.44])
=(-2.2297)/√(0.754478)(6.3536))
=(-2.2297)/√5.01999482
=(-2.2297)/2.24053449
=-0.9951643

The number is extremely close to -1, and we can thus establish the highly negative correlation between standard deviation of the dimensions of a face from the golden ratio and the beauty of a face. This means that the larger the deviation from the golden ratio, the less score it gets and vice-versa. This goes in line with the standard convention.
Relationship between deviation and average score

Now I will employ the Chi-square Goodness of fit test to check the reliability of the data using the equation below.
χ^2=∑▒〖(O-E)〗^2/E , where O= frequencies observed, and E= frequencies expected

The expected frequencies are located on the line of best fit which has equation y=0.3411x+3.7089. For each value of x, I can calculate the expected frequencies by substituting x into the equation of the line of best fit and use the formula to obtain a reduced chi-square value.