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The Golden Ratio

A Great Idea in Mathematics!
by

Amanda Rubalcava

on 12 May 2014

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Transcript of The Golden Ratio

The Golden Ratio and Beauty
The Golden Ratio
can be found in art and architecture all around the world!

So if the golden ratio is aesthetically pleasing, is human facial beauty defined by this divine proportion?
The Golden Ratio
In mathematics, we arrive at the golden ratio when the ratio of two quantities is the same as the ratio of their sum to the larger of the two quantities
This special ratio can be written as
The Golden Rectangle
And so with a golden rectangle, when we divide b/h, we always arrive at phi.
A Golden Rectangle
looks like:
So, what makes it special?
This ratio, which is also known as the divine proportion, gives an aesthetically pleasing height to width when applied to a rectangle.
So when a>b,
and is called
phi
A "golden rectangle" is one that is especially attractive, and whose b/h equals phi.
Phi can be defined as
So...
b/h = = a+b/a = a/b =1.6180339887.....
Recap:
To begin to list some, the ratios 8/5
and 13/8 all create golden rectangles and equal phi which approximates 1.618....
8
5
13
8
21
13
Let's examine the faces of some popular celebrities and see if the golden ratio holds true with their facial proportions!
Angelina Jolie
George Clooney
Denzel Washington
Enrique Iglesias
Beyonce
Kim Kardashian
We will also analyze celebrities of different race to add some variation.
Further....

The base and height of golden rectangles are also consecutive digits of the Fibonacci Sequence!












So, a golden rectangle is just a ratio of two digits of the Fibonacci Sequence.
To Measure: Using the Ondre Rulers computer application, I will measure the proportions of each face in centimeters.

I will measure to the nearest hundredth of a centimeter: (A) Top of the head to the chin; (B)Top of the head to the pupil; (C) Pupil to the tip of the nose; (D) Pupil to where lips meet; (E) Width of the nose; (F) Outside distance between the eyes; (G) Width of the head; (H) Hairline to the pupil; (I) Tip of the nose to the chin; (J) Lips to the chin; (K) Length of the lips; (L) Tip of the nose to the lips
Then,I will calculate these ratios for each celebrity:
A/G, B/D, I/J ,I/C, E/L, F/H, K/E
According to the Golden Ratio:

the most aesthetically appealing face would appear as such
Notice how the ratio of height to width (1.6/1) approximates the value of the golden ratio, which is 1.618..
After we calculate the ratios, our goal is to see which celebrity has the most perfect face according to math. We will determine this by seeing which face has the most golden ratios.
Angelina Jolie's Face Measurements
note: no matter the size of how big or small the picture of the celebrity is, since I am taking the ratios, the variations will not skew my results.
RATIOS:
A/G= 1.55
B/D=2.62
I/J=1.74
K/E=1.67
I/C= 2.33
E/L=0.88
F/H=1.97
Beyonce's Face Measurements:
RATIOS:
A/G=1.47
B/D=2.85
I/J=1.84
K/E=1.38
I/C=2.21
E/L=1.05
F/H=1.12
Kim Kardashian's Face Measurements:

RATIOS:
A/G=1.61
B/D=1.31
I/J=1.75
K/E=1.40
I/C=1.51
E/L=1.06
F/H=1.17
George Clooney's Face Measurements:

RATIOS:
A/G=1.62
B/D=1.37
I/J=1.6
K/E=1.35
I/C=1.69
E/L=1.29
F/H=1.21
Denzel Washington's Face Measurements:
RATIOS:
A/G=1.70
B/D=1.27
I/J=1.77
K/E=1.17
I/C=2.15
E/L=1.16
F/H=1.27

Enrique Iglesias' Face Measurements:
RATIOS:

A/G=1.68
B/D=1.62
I/J=1.62
K/E=1.29
I/C=1.92
E/L=1.28
F/H=1.15
Do celebrities who are considered beautiful all have faces with the golden ratio?
Big Questions:
Results:

Out of all six celebrities, Beyonce had no facial proportion that neared the Golden Ratio of 1.618 by at least a tenth.

Beyonce's closest proportion to the golden ratio was 1.47.
Results, Cont..
Out of all six celebrities, George Clooney and Enrique Iglesias had the highest amount of facial proportions nearing the golden ratio of 1.618.
George
1.62
1.6
1.69
Enrique
1.68
1.62
1.62
Results, Cont...
If we use the "ideal face" picture as the ultimate guide of the perfect face, George Clooney would come the closest. His length of face/ width of face is 1.62, where the Golden ratio is 1.618.
17.43 cm
10.76 cm
17.43/
10.76
= 1.62
Results, Cont.
The second face closest to the "ideal face" is Kim Kardashian. The ratio of the height of her face to the width of her face is 1.61
9.0 cm
14.53 cm
14.53/
9.0=
1.61
Conclusion
Five out of the six celebrities had facial proportions that neared the Golden Ratio.


While not all of the celebrities had faces that neared the Golden Ratio, I believe that there are enough to infer that the individuals who are considered "beautiful" are often the individuals who have facial proportions close to 1.618.
When constructing a Fibonacci sequence, you begin at 1 and 1, or 0 and 1, and depending on where you start, each subsequent number is the sum of the previous two
Works Cited
Burger, Edward B., and Michael P. Starbird.
The Heart of
Mathematics: An Invitation To Effective Thinking
. 4th ed.
Hoboken, NJ: J. Wiley & Sons, 2012. Print.
"Measuring Facial Perfection - The Golden Ratio."
Oprah.com.

Harpo Productions, Inc., 14 Aug. 2009. Web. 12 May 2014.
<http://www.oprah.com/oprahshow/Measuring-Facial-
Perfection-The-Golden-Ratio>.
Midler, Caryn. "Photo Special: All-Time Beautiful Men: Vote!"

People
. Time, Inc., 25 Apr. 2006. Web. 9 May 2014. <http:/
www.people.com/people/gallery/0%2C
%2C1184666_1184664_1055439%2C00.html#1055578>.
"Official Top 30 World's Most Beautiful Women of 2014."

Hollywood Buzz
. SpinMedia, 13 Dec. 2013. Web. 9 May
2014. <http://hollywoodbuzz.buzznet.com/user/
journal/17410302/official-top-30-worlds-most/>.
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