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# Golden ratio in beautiful faces

## Introduction

### Genesis

The golden ratio, also known as the number phi (φ), is an irrational number (a number which cannot be expressed as a fraction) which equals $$\frac{1+\sqrt5}{2}$$ (approximately 1.618). Two quantities represent the golden ratio if their sum divided by the larger quantity is equal to 1.618; it can be described in the form of the equation $$\frac{a+b}{a}$$

This mathematical concept is said to represent true beauty — that’s why it can also be called the golden or divine proportion. The research on the golden ratio dates back to Ancient Greece. The use of the letter ‘phi’ was begun by a Greek mathematician named Phidias who examined phi. It is very likely that the construction of the Parthenon was based upon the golden ratio.

Throughout the years, the golden ratio has been applied to many fields of man-made or natural origin. Nonetheless, the occurrence of the golden ratio in nature is highly debatable — there are probably more instances in which nature does not follow the golden ratio, even though it can be observed in some plants, skeletal systems, shells etc. The main industrial areas of interest utilizing the golden ratio include architecture, art, music and design

The golden ratio is supposed to represent proportions that appear most-pleasantly looking to the human eye. However, beauty is highly subjective. This made me think about the

relativity of human beauty and its link to proportion. I am generally really against sustaining beauty standards, so I often wonder where do they come from. The aim of my investigation is to examine the correlation of face proportions satisfying the golden ratio and the average rate of attractiveness (looks consistent with the prominent beauty standards) of a face.

### Golden ratio on the base of quadratic equation

The golden ratio can be described as - $$φ=\frac{a+b}{a}=\frac{a}{b}$$ where a = the longer part of the

segment and b = the shorter part of the segment. Thanks to this definition, the golden ratio can be expressed by a quadratic function.

$$φ =\frac{ a + b}{a} =\frac{a}{b}$$

$$φ =\frac{ a} {a} + \frac{b} {a}$$

$$φ = 1 + \frac{b} {a}$$

$$φ = 1 + \frac{1}{\frac{a}{b}}$$

$$φ = 1 +\frac {1} {φ}$$

$$φ2 = φ + 1$$

$$φ2 − φ − 1 = 0$$

Now, we can solve the equation using the quadratic formula -

$$x=\frac{-b\,±\sqrt{b^2-4ac}}{2a}=\frac{1±\sqrt{1-4(1)(-1)}}{2(1)}=\frac{1±\sqrt{5}}{2}=\frac{1+\sqrt{5}}{2}≈1.618\,and\frac{1-\sqrt{5}}{2}≈-0.618$$

The golden ratio is a comparison of two positive values which cannot be negative.
Therefore, the negative value has to be rejected.

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• ### Fibonacci sequence

The Fibonacci sequence1 is a set of numbers, starting from 0 or 1, where every next number represents the sum of the two proceeding numbers; e.g. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .... It can be represented in form of a mathematical equation, where = the term number ’n';

$$xn + 2 = xn + 1 + xn$$

The Fibonacci sequence is strictly related to the golden ratio. The ratio of any two successive numbers in the Fibonacci sequence will always be very close to the golden ratio (1.618). For example -

$$\frac{5} {3 }= 1.666; \frac{13} {8 }= 1.625; \frac{34} {21} = 1.619$$

Thanks to the golden ratio, the Fibonacci equation can be written with the use of phi (φ)

$$\frac{[φn − (−φ) −n]} {\sqrt{5}}$$

## Investigation

To start my investigation, I picked out 10 headshots of runways models. All of the pictures were taken in the same conditions (lighting, placing, angle etc.) For each photo, I measured their face proportions (vertical, transverse and external) and compared them to 2 the golden ratio. For each face, I calculated the average deviation of the vertical, transverse and external proportions from the golden ratio (1.618) . From many methods 3 available online, I chose the one, presented in the pictures below, that seemed the most comprehendible to me.

Figure 1 - Vertical proportion
Figure 2 - Transverse proportion
Figure 3 - External proportion

### Exemplary calculation of face A

Vertical proportion - forehead-nose (7.7 cm) vs nose-chin (4 cm) = 1.93 DEVIATION 1.93-1.618 = 0.31

Eyes-mouth (4cm) vs mouth-chin (2.1 cm) = 1.90 DEVIATION 1.90-1.618 = 0.29

Transverse proportion - mouth (3.1 cm) vs nose (2.2 cm) = 1.41 DEVIATION 1.41-1.618 = - 0.21

Forehead - cube root of (8.1 cm) = 2.01 DEVIATION 2.02 -1.61 8 = 0.40

Eyes - square root of (5.8 cm) = 2.41 DEVIATION 2.41-1.618 = 0.80

External proportion - length (13.5 cm) vs width (7.9 cm) = 1.71 DEVIATION 1.71-1.618 = 0.09

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### Video Course

• Figure 4 - Face A

I conducted an online survey among 36 participants. The interviewees were asked to rate the attractiveness of 10 faces of models on a scale from 0 to 4. The survey was Poland- based, so the beauty standards could remain culturally consistent.

Figure 5 - Table On Average Deviations
Figure 6 - Table On Number Of Particular Scores Given

Then, the average scores of face attractiveness from the survey were compared with the

$$r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}}$$

summations of average deviations. To investigate the correlation, I had to utilize the equation of correlation which allowed be to calculate the correlation coefficient.

### Exemplary calculation of average score (Face A)

(0 × 1) + (1 × 4) + (2 × 14) + (3 × 12) + (4 × 5) = 2.44

Hence, I calculated the x y, x2 and y2 values for every x and y value. Then I calculated the summations of every value, in order to insert them into the formula.

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### Video Course

• Figure 7 - Table On Correlation Coefficient Formula Values

$$r = \frac{10(14.39) - (6.78)(22.78)}{\sqrt{\bigg[10 \cdot (5.05) - (6.79)^2\bigg][10 \cdot (56.34) - (22.78)^2]}} = - \,0.362665$$

The correlation coefficient suggests that the correlation between the two variables (deviation from the golden ratio and average score) is negative and weak. It would appear that the value of the deviation decreases with respect to the value of the average scores, but there is a low likelihood of a relationship between variables. After inserting the values into my GDC, I could obtain the equation of line regression, which would also act as the line of best fit in a scatter diagram. Nevertheless, it does not prove to be useful in case of such a weak correlation.

Figuer 8 - Scatter Diagram
Figure 9 -

As we can see, the Pearson’s correlation coefficient calculated by GDC is closely similar to the coefficient calculated by hand.Chi-squared test was applied to investigate if there was any dependency between the variables, the average deviation of the face compared to the golden ratio and the attractiveness scores from the survey. I grouped the faces for clarity based on similar average deviation compared to the golden ratio - 0 - 0.6 (Faces B, C, F, H); 0.61 - 0.8 (Faces D, E, A, J) and >0.81 (Faces G, I)

H0 - The calculated measurement standard deviation from the golden ratio and the point- point-expressed attractiveness survey score are independent.

H1 - The calculated measurement standard deviation from the golden ratio and the point- point-expressed attractiveness survey score are dependent.

Figure 10 - Chi-Squared Test Matrices
Figure 11 - Observed values
Figure 12 - Expected values

In a 3x5 table, we can calculate the degrees of freedom by utilizing the formula - (r - 1)(c - 1) , where ‘ ’ represents the number of rows and ‘ ’ represents the number of columns.

The result of the equation - (3 -1)(5 -1) allows us to search for the critical value in a critical value table5. Critical value is a point that determines whether the null hypothesis (The variables are not dependent on each other) should be rejected or not. As the Chi- squared test value is greater than the critical value (15.51), we can assume that one variable causes the other

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• ## Limitations

The interviewed sample consisted of 36 participants. This is not enough to include all definitions of beauty and their origin. The study may be biased because of the sample size. The study cannot be cross-culturally universal, because it only investigated beauty standards in the eyes of Polish people.

If more time had been devoted to the investigation, the study would’ve been more thorough and relevant. The time limitation doesn’t allow for further interviews, making it impossible to obtain the most accurate data.

The survey was conducted mainly between high school students. The sample is not diverse in age, ranging from 16 to 20 years old. This may’ve influenced the ratings, as beauty standards has changed throughout time.

## Conclusion

There is a weak, negative correlation suggested by the correlation coefficient and chi- squared test between face proportions satisfying the golden ratio and average scores of attractiveness. This may imply that even though people have their own preferences, face proportions play a small part in rating attractiveness. The lower the deviation of the proportion ratio from the golden ratio, the higher the average score. The result of the Chi- squared test proved that we can assume there is a relationship between the variables. The smaller deviation from the golden ratio results in a greater average score of attractiveness.The golden ratio can also be presented in a form of a quadratic function. It is closely connected to the Fibonacci sequence.

## Bibliography

• Correlation coefficient - Simple definition, formula, easy calculation steps. (2021, February 03). Retrieved November 26, 2020, from https://www.statisticshowto.com/ probability-and-statistics/correlation-coefficient-formula/
• Critical values of the Chi-Square Distribution. (n.d.). Retrieved February 26, 2021, from https://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm
• Livio, M. (2008). The Golden Ratio: The story of Phi, the world's most astonishing number. New York: Broadway Books.
• Schneider, R. (2016). Fibonacci numbers and the golden ratio. arXiv: History and Overview.

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• ## Appendix A

### Face A

Vertical proportion - forehead-nose (7.7 cm) vs nose-chin (4 cm) = 1.93 DEVIATION 1.93 -1.618 = 0.31 Eyes-mouth (4cm) vs mouth - chin (2.1 cm) = 1.90 DEVIATION 1.90-1.618 = 0.29

Transverse proportion - mouth (3.1 cm) vs nose (2.2 cm) = 1.41 DEVIATION 1.41-1.618 = - 0.21 Forehead (8.1 cm) cubed = 2.01 DEVIATION 2.02-1.618 = 0.40 Eyes (5.8 cm) squared = 2.41 DEVIATION 2.41-1.618 = 0.80

External proportion - length (13.5 cm) vs width (7.9 cm)= 1.71 DEVIATION 1.71-1.618 = 0.09

### Face B

Vertical proportion - forehead-nose (6.7 cm) vs nose-chin (3.6 cm) =1.86 DEVIATION 1.86 -1.618 = 0.24 Eyes-mouth (4 cm) vs mouth-chin (2.1 cm)=1.90 DEVIATION 1.90 -1.618 = 0.29

Transverse proportion - mouth (2.8 cm) vs nose (2.1 cm) =1.33 DEVIATION = 1.33-1.618 = -0.29 Forehead (7.8 cm) cubed = 1.98 DEVIATION 1.98 -1.618 = 0.36 Eyes (5.4 cm) squared = 2.32 DEVIATION 2.32-1.618 = 0.70

External proportion - length (13.3 cm) vs width (8.1 cm) = 1.64 DEVIATION = 1.64-1.618=0.02

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• ### Face C

Vertical proportion - forehead-nose (7.3 cm) vs nose-chin (4.3 cm) =1.70 DEVIATION 1.70 -1.618 = 0.08 Eyes-mouth (4.4 cm) vs mouth - chin (2.5 cm) = 1.76 DEVIATION = 0.14

Transverse proportion - mouth (2.8 cm) vs nose (2.1 cm) =1.33 DEVIATION 1.33-1.618 = - 0.29 Forehead (8.3 cm) cubed = 2.02 DEVIATION 2.02 -1.618 = 0.34 Eyes (6.2 cm) squared = 2.49 DEVIATION 2.49 -1.618 = 0.87

External proportion - length (14.5 cm) vs width (8.7 cm) = 1.66 DEVIATION 1.66-1.618 = 0.04

### Face D

Vertical proportion - forehead-nose (7.3 cm) vs nose - chin (3.8 cm) = 1.92 DEVIATION 1.91-1.618 = 0.30 Eyes - mouth (4 cm) vs mouth - chin (2.3 cm) = 1.74 DEVIATION 1.74 -1.618 = 0.12

Transverse proportion - mouth (2.9 cm) vs nose (2.1 cm) = 1.38 DEVIATION 1.38-1.618 = - 0.24 Forehead (8.2 cm) cubed = 2.01 DEVIATION 2.01-1.618 = 0.39 Eyes (5.8 cm) squared = 2.41 DEVIATION 2.41-1.618 = 0.79

External proportion - length (14.3 cm) vs width (8.3 cm) =1.71 DEVIATION 1.71-1.618 = 0.09

### Face E

Vertical proportion - forehead-nose (8 cm) vs nose-chin (4.2 cm) = 1.90 DEVIATION = 1.90 -1.618 = 0.28 Eyes-mouth (4.1 cm) vs mouth-chin (2.6 cm) = 1.58 DEVIATION = - 0.04

Transverse proportion - mouth (3.1 cm) vs nose (2.4 cm) = 1.29 DEVIATION = 1.29 -1.618 = -0.33 Forehead (8.5 cm) cubed = 2.04 DEVIATION 2.04 -1.618 = 0.42 Eyes (5.9 cm) squared = 2.43 DEVIATION 2.43 -1.61 8= 0.81

External proportion - length (14.4 cm) vs width (8.2 cm) = 1.76 DEVIATION 1.76-1.618 = 0.14

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• ### Face F

Vertical proportion - forehead - nose (7.5 cm) vs nose - chin (3.4 cm) = 2.21 DEVIATION = 2.21-1.618 = 0.59 Eyes - mouth (3.8 cm) vs mouth - chin (2.3 cm) = 1.65 DEVIATION = 0.03

Transverse proportion - mouth (2.6 cm) vs nose (2.1 cm) = 1.24 DEVIATION = 1.29 -1.618 = - 0.38 Forehead (7.4 cm) cubed = 1.95 DEVIATION 1.95-1.618 = 0.33 Eyes (5.2 cm) squared = 2.28 DEVIATION 2.28 -1.618 = 0.66

External proportion - length (13 cm) vs width (8.2 cm) = 1.59 DEVIATION 1.59 -1.618 = - 0.02

### Face G

Vertical proportion - forehead - nose (8.9 cm) vs nose  -chin (4.2 cm) = 2.12 DEVIATION = 2.12-1.618 = 0.50 Eyes - mouth (4.9 cm) vs mouth - chin (2.5 cm) = 1.96 DEVIATION = 0.34

Transverse proportion - mouth (3.1 cm) vs nose (1.8 cm) = 1.72 DEVIATION = 1.72-1.618 = 0.10 Forehead (8.2 cm) cubed = 2.02 DEVIATION 2.02-1.618 = 0.40 Eyes (6.9 cm) squared = 2.62 DEVIATION 2.62 -1.618 =1.01

External proportion - length (16.1 cm) vs width (8.6 cm) = 1.87 DEVIATION 1.76 -1.618 = 0.25

Vertical proportion - forehead - nose (8.1cm) vs nose - chin (4.4 cm) = 1.84 DEVIATION = 1.84 -1.618 = 0.22 Eyes - mouth (4.3 cm) vs mouth-chin (2.9 cm) = 1.4 DEVIATION = - 0.22

Transverse proportion - mouth (2.9 cm) vs nose (2.3cm) = 1.26 DEVIATION = 1.26 -1.618 = - 0.36 Forehead (8.5 cm) cubed = 2.04 DEVIATION 2.04 -1.618 = 0.42 Eyes (6.1 cm) squared = 2.47 DEVIATION 2.47-1.618 = 0.85

External proportion - length (15.7 cm) vs width (9.1 cm) = 1.73 DEVIATION 1.73-1.618 = 0.11

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• ### Face I

Vertical proportion - forehead - nose (7 cm) vs nose - chin (3.8 cm) = 1.84 DEVIATION = 1.84 -1.618 = 0.22 Eyes - mouth (4.3 cm) vs mouth - chin (2.2 cm) = 1.95 DEVIATION = 1.95 -1.618 = 0.34

Transverse proportion - mouth (3.2 cm) vs nose (2.3cm) = 1.39 DEVIATION = 1.39 -1.618 = - 0.23 Forehead (7.7 cm) cubed = 1.97 DEVIATION 2.04-1.618 = 0.36 Eyes (6.1 cm) squared = 2.47 DEVIATION 2.47-1.618 = 0.85

External proportion - length (15.3 cm) vs width (8.1 cm) = 1.88 DEVIATION 1.88 -1.618 = 0.27

### Face J

Vertical proportion - forehead - nose (8.1 cm) vs nose - chin (4.2 cm) = 1.93 DEVIATION = 1.93 -1.618 = 0.31 Eyes-mouth (4.4 cm) vs mouth-chin (2.9 cm) = 1.52 DEVIATION = 1.52 -1.618  =- 0.10

Transverse proportion - mouth (3.2 cm) vs nose (2.1cm) = 1.52 DEVIATION = 1.52-1.618 = - 0.09 Forehead (9.8 cm) cubed = 2.14 DEVIATION 2.14-1.618 = 0.52 Eyes (6.3 cm) squared = 2.51 DEVIATION 2.51-1.618 = 0.89

External proportion - length (15.8 cm) vs width (9.4 cm) = 1.68 DEVIATION 1.68-1.618 = 0.06

### Appendix B

Figure 13 - Table On

Figure 14 -

Figure 15 -