Final MATH IA for ib documents which scored 7 PDF

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Summary

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see Fig. 7.23). Show that:
(i) ∆AMC≅∆BMD
(ii) ∠ DBC is a right angle. (iii) ∆ DBC ≅ ∆ ACB
1 (iv)CM= 2 AB...

Description

INTERNATIONAL BACCALAUREATE MATHEMATICS INTERNAL ASSESSMENT

Research Question- How do the best-looking celebrities in the world correlate with the Golden ratio method identifying beauty?

S.NO

TITLE

PAGE NO.

1.

Introduction

2

2.

Statement of task

2

3.

Plan of investigation

2

4.

Numerical data

3

5.

Raw Data

4

6.

Data for women from magazine

4

7.

Data for women based on golden ratio

5

8.

Data for men from magazine

6

9.

Math calculations

8

10.

Correlation for women data

8

11.

Correlation for men data

10

12.

Limitations

13

13.

Conclusion

13

14.

Citations

14

INDEX

1

Introduction/Rationale: The golden ratio is a method that was created by Phidias who was a Greek sculptor and mathematician around 500 BC TO 432 BC. It is a method used to map out the perfect proportions of natural beings and objects. This was commonly used as a method of the identification of beauty in a person. It was later popularized by Renaissance artist’s like Leonardo Da Vinci to map out their masterpieces.

Statement of the task: The main purpose of my mathematics studies internal assessment is to decipher whether the golden ratio’s ideals of beauty correlates with the ideals of beauty that celebrities possess. It will be an analysis of the golden ratio of celebrities that are considered to be the best-looking people in the world from a list that is curated by a magazine. I will then find out how they correlate with the concept of the golden ratio. I will also create a list ranking the celebrities from the highest golden ratio to the lowest golden ratio. This method states to confirm if archaic standards of beauty like the golden ratio still stands the true test of time. I believe that beauty lies in the eyes of the beholder and is a highly subjective concept that is tough to quantify correctly. Every single person in this world has its endearing qualities that appeal to a certain number of people. For this IA, my data is focused on celebrities that have a higher percentage of people who find them appealing. 2

Plan of Investigation: The data has been collected from a website, www.anaface.com 1that derives the golden ratio from finding out distances between multiple areas on a face. The perfect ratio of the length of the face is 1.618 times that of the breadth of the face. Similarly, the length from the hairline to the pupil should be the same from the pupil to the bottom of the nose. This should also be the case from the bottom of the nose to the bottom of the chin. The golden ratio also compares the symmetry of the face. The distance between the two eyes should be the exact length of one eye. The length of the ear should correlate to the length of the nose. If such numbers are exact, the person is scientifically considered more beautiful. The golden ratio uses phi proportions to identify the numbers. The website then automatically converts the golden ratio of a person into a score out of ten. The analysis of the faces of women celebrities and men celebrities have been conducted differently. The golden ratio of two genders is correlated with the data from the magazines. The golden ratio is calculated based on the following proportions… 1. The length of the face to the breadth of the face 2. The distance between the eyes to the distance between both eyes 3. The length of the ears to that of the nose 4. The breadth of the nose to the distance between the eyes 5. The distance between the hairline to the pupil to that of the bottom to the bottom of the nose to the bottom of the chin.

This data is analysed when a picture is uploaded on the website and 17 points on the face are highlighted like shown below.

1 “Golden Ratio Calculator.” Calculate the Symmetry of Your Face Using Anaface, www.anaface.com/. 3

Numerical values: In both the data for men and women the term x, given in the table denotes the values that have been collected from a magazine. This data is not certified it is subjective and based on the opinion of the magazine. The term y denotes the data that is calculated through the method of the golden ratio.

RAW DATA: Data for women celebritiesThe data below is taken from the web magazine and the data is arranged according to the ranking in the magazine. S.NO 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 4

CELEBRITY Megan Fox Jessica Alba Scarlett Johansson Charlize Theron Kate Upton Natalie Portman Mila Kunis Adriana Lima Gabrielle Union Selena Gomez Emma Watson Kate Beckinsale Jennifer Aniston Rihanna Olivia Wilde Keira Knightly Blake Lively

GOLDEN RATIO 7.37 7.25 8.09 7.46 7.34 7.77 7.85 7.9 7.11 8.53 6.87 7.32 7.50 7.83 7.47 7.40 7.03

18 19 20 21 22 23 24 25

Jennifer Lawrence Rachel McAdams Hayden Pettiere Salma Hayek Jessica Lucas Kristen Bell Shakira Katy Perry

7.2 7.29 8.09 8.15 7.37 7.79 7.61 7.68

Women List According to Symmetry of golden ratio The data below is the magazine data presented from the highest to lowest golden ratio, pertaining to each celebrity. S.NO 1. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

5

CELEBRITY Selena Gomez Salma Hayek Scarlett Johansson Hayden Panettiere Adriana Lima Mila Kunis Rihanna Kristen Bell Natalie Portman Katy Perry Shakira Jennifer Anniston Olivia Wilde Charlize Theron Keira Knightley Megan Fox Jessica Lucas Kate Upton Kate Beckinsale Rachel McAdams Jessica Alba Jennifer Lawrence Gabrielle Union Blake Lively Emma Watson

GOLDEN RATIO 8.53 8.15 8.09 8.04 7.90 7.85 7.83 7.79 7.77 7.68 7.61 7.50 7.47 7.46 7.40 7.37 7.37 7.34 7.32 7.29 7.25 7.20 7.11 7.03 6.87

Men’s records- according to the magazine data The data below is taken from the web magazine and the data is arranged according to the ranking in the magazine. S.NO 1. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

CELEBRITY Jamie Dornan Idris Elba Henry Cavill David Beckham Ryan Gosling Ryan Reynolds Bradley Cooper Jake Gyllenhall Robert Pattinson Pharrell Williams Tom Hardy Johnny Depp Hugh Jackman Zac Efron George Clooney Harry Judd Garry Barlow Gerard Butler James Franco Taylor Lautner Brad Pitt Orlando Bloom Justin Timberlake Olly Murs Tinie Tempah

GOLDEN RATIO 8.73 8.25 7.28 8.2 7.61 6.59 7.71 7.89 8.43 7.03 8.65 7.86 7.27 7.26 6.51 8.51 7.19 8.49 8.64 7.6 7.38 6.3 7.18 7.69 7.97

Men’s records- according to symmetry of golden ratio The data below is taken from the web magazine and the data is arranged according to the ranking in the magazine. S.NO 6

CELEBRITIES

GOLDEN RATIO

1. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Jamie Dornan Tom Hardy James Franco Harry Judd Gerard Butler Robert Pattinson Idris Elba David Beckham Tinie Tempah Jake Gyllenhall Johnny Depp Bradley Cooper Olly Murs Ryan Gosling Taylor Lautner Brad Pitt Henry Cavill Hugh Jackman Zac Efron Gary Barlow Justin Timberlake Pharrell Williams Ryan Reynolds George Clooney Orlando Bloom

8.73 8.65 8.64 8.51 8.49 8.43 8.25 8.2 7.97 7.89 7.86 7.71 7.69 7.61 7.6 7.38 7.28 7.27 7.26 7.19 7.18 7.03 6.59 6.51 6.3

MATH CALCULATIONS

Women’s DataMean= (sum of all values/ number of total values) = 189.22/25=7.5688

x- The values in the X column are the values taken from the magazine that are arranged on the basis on the hierarchy given in the magazine y- The values in the Y column are those that are arranged from highest to lowest, based on the golden ratio. x 7

y

x -

y -

( x - x

( x−x )2

( y− ´y )2

´y

x

)( y ´y )

1 2 3 4 5 6 7 8 9 0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Σ

7.37 7.25 8.09 7.46 7.34 7.77 7.85 7.90 7.11 8.53 6.87 7.32 7.50 7.83 7.47 7.40 7.03 7.20 7.29 8.04 8.15 7.37 7.79 7.61 7.68 189.22

8.53 8.15 8.09 8.04 7.90 7.85 7.83 7.79 7.77 7.68 7.61 7.50 7.47 7.46 7.40 7.37 7.37 7.34 7.32 7.29 7.25 7.20 7.11 7.03 6.87 189.22

0.9612 0.5812 0.5212 0.47 0.3312 0.2812 0.2612 0.2212 0.2012 0.1112 0.0412 -0.0688 -0.10 -0.1088 -0.1688 -0.1988 -0.1988 -0.2288 -0.2488 -0.2788 -0.3188 -0.3688 -0.4588 -0.5388 -0.6988

Correlation Coefficient=

0.9612 0.5812 0.5212 0.47 0.3312 0.2812 0.2612 0.2212 0.2012 0.1112 0.0412 -0.0688 -0.10 -0.1088 -0.1688 -0.1988 -0.1988 -0.2288 -0.2488 -0.2788 -0.3188 -0.3688 -0.4588 -0.5388 -0.6988

r=

-0.1910866 -0.1852866 0.27164944 -0.0512666 -0.0757786 0.05657744 0.07344944 0.07326144 -0.0923106 0.10688544 -0.0287906 0.01711744 0.00679744 -0.0284186 0.01667744 0.03355744 0.10711344 0.08438144 0.06936544 -0.1313706 -0.1852866 0.07331744 -0.1014866 -0.0221986 -0.0777066 -0.180836

0.03952144 0.10163344 0.27164944 0.01183744 0.05234944 0.04048144 0.07907344 0.10969344 0.21049744 0.92390544 0.48832144 0.06190144 0.00473344 0.06822544 0.00976144 0.02849344 0.29030544 0.13601344 0.07772944 0.22202944 0.33779344 0.03952144 0.04892944 0.00169744 0.01236544 3.668464

0.92390544 0.33779344 0.27164944 0.22202944 0.10969344 0.07907344 0.06822544 0.04892944 0.04048144 0.01236544 0.00169744 0.00473344 0.00976144 0.01183744 0.02849344 0.03952144 0.03952144 0.05234944 0.06190144 0.07772944 0.10163344 0.13601344 0.21049744 0.29030544 0.48832144 3.668464

∑ ( x− x´ ) ( y− ´y ) √ ∑ ( x−´x) 2−∑ ( y −´y )2

Therefore, the correlation is = -0.1430562 i.e. there is weak negative correlation between the best-looking women celebrities in the world and the golden ratio method of analysing beauty. This weak negative correlation denotes what the magazine believes to be the most beautiful women celebrities does not at all coincide with the data. In fact, the women that the magazine considers to be less good looking have a higher golden ratio.

8

Chart Title 9 8 7 6 5 4 3 2 1 0 Data according to magazine hyrarchy

Data according to hyerarchy of golden ratio

The above graph compares the data that is extracted from the online magazine to the data of the golden ratio of female celebrities. The above graph compares the data that is extracted from the online magazine to the data of the golden ratio of female celebrities. This data shows the consistent irregularities in both the calculations of beauty. The highest hierarchy of the celebrity in the magazine does not correlate with the data in the golden ratio, it is shown in a much lower spot on the line of the golden ratio. The lowest hierarchy of the celebrity given in the magazine is much higher compared to the golden ratio. Data for Male Celebrities

Mean= (sum of all values/ number of total values) = 192.22/25 7.688

x- The values in the X column are the values taken from the magazine that are arranged on the basis on the hierarchy given in the magazine y- The values in the Y column are those that are arranged from highest to lowest, based on the golden ratio.

x

y

x -

y -

( x−´x

´y

)( y− ´y

´x

) 9

(x−x )2

( y - ´y )2

1

1.041

9.08967

8.73

8.73

2 0.561

2 0.961

6

76.2129

1.08409744

8.25

8.65

2 -

2

7.9299

68.0625

0.92390544

0.408

0.951

6.92473

8 0.51 -

2 0.82

6 6.73384

52.9984 67.24

0.90478144 0.67436944

0.078

0.801

6.09713

8 -

2

2

57.9121

0.64192144

1.098

0.741

4.88450

2 3

4 5

1.041

7.28 8.2

7.61

8.64 8.51

8.49

6

6.59

8.43

8 0.021

2 0.561

8 4.32685

43.4281

0.54937744

7.71

8.25

2 0.201

2

2 4.03336

59.4441

0.31494544

7.89

8.2

2 0.741

0.5112 8 0.281 2.37051

62.2521

0.26132544

8.43

7.97

2 -

2

6

71.0649

0.07907344

0.658

0.201

1.41443

7 8 9 0

7.03

7.89

8 0.961

2 0.171

6

49.4209

0.04048144

8.65

7.86

2 0.171

2 0.021

1.48088 0.16663

74.8225

0.02930944

2 1

7.86

7.71

2

2

2 0.00872

61.7796

0.00044944

3 1

7.27

7.69

-0.42 -

0.00 -

4 -

52.8529

1.44E-06

0.428

0.078

0.57208

8 -

8 -

8 -

52.7076

0.00620944

1 5

1.178

0.088

0.57808

8

8 -

8 -

42.3801

0.00788544

0.821

0.308

2.62788

2 -

8 -

8 -

72.4201

0.09535744

1 7

0.498

0.408

2.93927

8

8

2

51.6961

0.16711744

11 1

4 7.26

6.51

7.61

7.6

1 6 8.51

7.19 10

7.38

7.28

1

-

-

0.801

0.418

3.55561

2

8 -

2 -

0.951

0.428

3.70483

2

2 -

8 -

0

0.088

0.498

2

8 -

1

8 8.49

7.27

72.0801

0.17539344

2

74.6496

0.18386944

8 -

-3.79088 -

57.76

0.24880144

0.308

0.508

3.75494

8 -

8 -

4

54.4644

0.25887744

2 2

1.388

0.658

8 -

8 -

-4.15044 -

39.69

0.43401744

2 3

0.508

1.098

7.88938

8

8 -

4 -

51.5524

1.20736144

0.001

1.178

9.06497

2

8 -

2

0.000001

1.38956944

0.281

1.388

-

2

8

11.06873 63.5209 1.92876544 1.76406 11.6072636 11.607264

1 9 8.64

7.6

7.38

6.3

7.18

7.26

7.19

7.18

7.03

6.59

2 4 7.69

6.51

2 5 Σ

7.97 192.2

6.3 192.2

2

2

Correlation Coefficient=

4

r=

∑ ( x− x´ ) ( y− ´y ) √ ∑ ( x−´x )2−∑ ( y −´y ) 2

Therefore, the correlation is= 0.15197932 e.g. there is weak positive correlation between the best-looking men celebrities in the world and the golden ratio method of analysing beauty. This positive weak correlation states that what the magazine thinks is handsome has a slight correlation with the golden ratio of the same group of celebrities. 11

Comparative data of magazine and golden ratio hierarchy for male celebrities 10 9 8 7 6 5 4 3 2 1 0 Category 1 Data according to magazine hierachy

Data according to the hierarchy of golden ratio

The above graph compares the data that is extracted from the online magazine to the data of the golden ratio of male celebrities. This data shows the consistent irregularities in both the calculations of beauty. Even though the highest ranked person in both categories of golden ratio and magazine collected data are the same, there is a strong disparity in the lowest ranked person.

12

Limitations: One major limitation is that the data that I collected from an internet magazine is largely based on the opinion of the editors of that magazine and not data from a poll. There data is influenced by that of its readers and from opinions seen on social media. Beauty, as a concept is extremely subjective, and the data may differ from person to person.

Another limitation is that the process of golden ratio that I have collected is obtained through another website that has the possibility of being inaccurate.

Conclusion- For both men and women, post data collection, the values of correlation denote little to no association between the list from a magazine and the data from the golden ratio. For the women celebrities, there is weak negative correlation between the data from the magazine and the golden ratio of the data. This denotes that as the golden ratio of the data increases, the women are considered less beautiful or good-looking. Even though the data is taken of the supposed ‘best looking’ people in the world, the golden ratio negatively correlates with the data. For the men celebrities, there is a weak positive correlation between the data from the magazine and the golden ratio of the data. This denotes that as the golden ratio of the data increases, the male celebrities are considered to be more handsome or good-looking. This is a contrast to the women celebrities that have a negative correlation with the data and the golden ratio. Comparing the mean ratios of both men and women, we can also tell that the average golden ratio of men is higher than that of women

My study proves that beauty is not scientific concept and cannot be denoted through scientific means. It is a subjective idea and cannot be distinguished through mathematical or scientific means. For one set of data there is a negative correlation but for one set of data there is a positive correlation. This denotes a lack of consistency in the opinion of the magazines and the golden ratio.

13

CITATIONS

Chouhan, Monica. “The Top 25 Sexiest Men In The World.” Heart, 29 Aug. 2018, www.heart.co.uk/showbiz/top-sexiest-men/.

“Find the Golden Ratio of Faces.” Anaface- The Golden- Ratio Finder , www.anaface.com/. “Golden Ratio.” Math Is Fun, www.mathsisfun.com/numbers/golden-ratio.html.

“Golden Ratio.” From Wolfram MathWorld, mathworld.wolfram.com/GoldenRatio.html.

“Golden Ratio Calculator.” Calculate the Symmetry of Your Face Using Anaface, p3nlhclust404.shr.prod.phx3.secureserver.net/SharedContent/redirect_0.html.

Wallin, Emmy. “The 25 Hottest Women in the World....