golden ratio project phase 2

7/31/2019 Golden Ratio Project Phase 2

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Running head: THE GOLDEN RATIO 1

The Golden Ratio

Katie Votaw

Stats-1510

Taft College


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THE GOLDEN RATIO 2

Abstract

The purpose of this experiment is to determine if there is validity in the idea of the Golden Ratio

as it relates to the measurements of the average human body. The golden ratio states that a+b is

to a as a is to b and is based off of the Fibonacci sequence that is built by adding the two

previous numbers together to get the next number (Harris, n.d.). What I wanted to find out was if

the Golden ratio has a true presence within the real human body or if it is only present in the

idealized, possibly even mythic, human body. Prior to conducting my research my hypothesis

was that the golden ratio exists only as an idealization of the human body, but does not apply to

real human proportions. I believe that I will find that each person varies in the proportions of

their body parts measured and that none will meet the standard, 1.618, of The Golden Ratio. In

order to test this idea I measured 30 of my co-workers, family, and friends of a range of different

heights and builds and calculated the resulting ratios to determine if they fit the mold of the

Golden Ratio 1/1.618. The TC-Stats Ipad application was used to construct graphical displays,

frequency distribution, and summary statistics tables in order to assist us in determining any

differences between the three sets of data. After determining that our data sets were not

normally distributed we chose to use the Kruskal-Wallis technique to determine if there were any

differences. The technique concluded that there was a difference in the data sets which allowed

us to then look at the median of each set of data which brought us to the conclusion that each

data set was completely different from each of the others. Some were similar in the proportions

of measurements of certain body parts, but not one person had The Golden Ratio proportion on

all or more than one of their corresponding body part measurements.


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THE GOLDEN RATIO 3

The Golden Ratio

In this report I will discuss the idea of the Golden Ratio as it applies to the real human

body. The Golden Ratio is a mathematical idea first noticed by ancient Greeks in geometry.

They found that objects that followed the idea of this Golden Ratio were the objects that were

most pleasing to the eye. These objects included items in nature such as the shapes of flowers

and sea shells and extend to objects like a piece of paper or a window. After discovering this

pattern found in many objects the Greeks began to use this ratio in the architecture of their

buildings, to create their artwork, and even to compose their music (Parveen, n.d.). The numeric

value for this ratio is called Phi which equals approximately 1.618 and its dimensions have been

used to create aesthetically appealing objects ever since its discovery (The Golden Ratio, n.d.).

I believe that there is a possibility that if Man could build a human being that its body

measurements would calculate to the proportions of the golden ratio, however I do not feel that

on this planet we have many people, if any, that fit into this very strict mold. We are all made

differently and while you can create a rectangular object whose width to height ratio equals

1/1.618 and whose width plus the height divided by the height equals 1.618 we are not capable

of intentionally creating humans measurements to meet these specifications.

Method

Materials

I used a tape measure that goes up to 120 inches to measure the necessary parts of each

participants body. There were a total of 30 people who volunteered to participate in my

research. After the data was collected I used TC-Stats to create the visual displays of the data, to

calculate the golden ratio and see if it applied to any of my participant s measurements, and to


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THE GOLDEN RATIO 4

apply the Kruskal-Wallis technique in order to determine if there were any significant

differences between the multiple sets of data.

Measures

In this experiment I wanted to determine if any of my participants measurements fit the

ideal proportions given to us by The Golden Ratio. I also wanted to see if there were several

people whose measurements fell into the golden ratio category, a few people, or no one at all. In

order to do this I used the Kruskal-Wallis technique. The Kruskal-Wallis technique tells us if

there are any significant differences between our sets of data. The limitation to the Kruskal-

Wallis technique is that it cannot tell us which set of data is different from the others or if

multiple data sets are different. In order to do this we look at the medians of each data set to see

if we can see a significant difference in any of the data.

Participants

In this experiment there were 30 voluntary participants including 17 women and 13 men. The

mean age of this group is 32.65. Each volunteer had 18 different measurements taken from their

arms, legs, torso, and head and each measurement was repeated and compared to other

measurements on the same participant in order to account for accuracy.

Procedure

In order to increase the chance of finding someone who closely fits the measurements of

the golden ratio I wanted to have a wide variety of body types represented. In order to do this I

made sure I measured as many people as I could. I took a group of 30 people from work, my

family, and group of friends and measured different sections of their body with a measuring tape.

I repeated the measurements several times in order to make sure I had an accurate measurement

of everything. On each person I measured the distance from their naval to the end of their foot,


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THE GOLDEN RATIO 5

their full height, the distance from their finger-tip to their elbow, from their wrist to their elbow,

from their shoulder-line to the top of their head, the length of their head, the distance from their

naval to the top of their head, from their naval to their knee, from their knee to the end of their

foot, the length of the first two sections of their index finger, the length of their index finger, the

length of the middle finger, the length of the pinky finger, the length of their face, and lastly the

width of their face. I then used TC-Stats to calculate the ratio for each measurement for each

person in order to see if it fit the golden ratio formula. I also used TC-Stats to create visual

displays of all of my data including normal plots to determine if any of the data was in fact

normal. After concluding that the data was not normal I used the Kruskal-Wallis technique to

determine if there were any significant differences in my data sets.

Results

Before conducting this experiment my hypothesis was that there would not be any person

who fit the ideal proportions of the golden ratio 1/1.618, and that each persons measurements

would be significantly different from the other participants.

After applying the Kruskal-Wallis technique with to our data we found significant

evidence to suggest that there are significant differences in our data sets. After applying the

golden ratio formula to our measurements we have also concluded that while some proportions

of some measurements of participants seemed close to the golden ratio of 1.618 none were close

enough to suggest that the golden ratio actually exists in a real human body. Therefore my

hypothesis that there would be significant differences in each participants measurement

proportions, and that the real human body does not conform to the ideals of the golden ratio was

correct.

Discussion


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THE GOLDEN RATIO 6

Based on my research I was able to determine there were significant differences in each

of the data sets compared to each of the others by using the Kruskal-Wallis technique and then

comparing the medians of each of the data sets. I then calculated the ratios of each of the

measurements of each person. In order to fit the golden ratio each measurement when compared

to its corresponding measurement should result in the formula of a+b/a equaling 1.618. After

viewing the results of my calculations I concluded that none of the measurements matched the

idealistic Golden Ratio 1.618. According to my research there is significant evidence to suggest

that the golden ratio does not exist in the real human body, that it may only be the idealistic

measurements.


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THE GOLDEN RATIO 7

Figure 1: Raw Data and Labels

Participant # A B C D E F G H I J K L M N O

1 73 41 16 11 13 10 32 19 22 1.7 3.3 3.5 2.6 9 7

2 60 39 15 9 10 9 21 19 20 1.5 3.2 3.4 2.4 10 8

3 61 40 16 10 11 8.5 21 17 23 1.4 3.1 3.4 2.3 8 7

4 65 37 16 9 13 9 28 16 21 1.4 3 3.4 2 9 8

5 69 39 15 8 12 10 30 17 22 1.5 3.2 3.5 2.2 9 7.5

6 62 39 15 9 11 10 23 18 21 1.7 3 3.5 2.5 9.5 8

7 67 40 17 10 12 11 27 19 21 1.5 3 3.4 2.5 10 8

8 68 41 14

.5

8 13 9.5 27 22 19 1.4 3.1 3.2

5

2.3 8 7

9 63 38 15

.5

9 11 10 25 17 21 1.4 3.3 3.5 2.4 9 8

10 65 37 16 8 12

.5

10 28 17 20 1.5 3.2 3.4 2.4 9 7

11 65 38 14 10 13 11 27 18 20 1.4 3.1 3.5 2.2 10 8.5

12 70 43 16 11 13

.5

10 27 20 23 1.5 3.2

5

3.5 2.5 9 8

13 66 41 16 9 12 10 25 18 23 1.4 3.2 3.4

5

2.3 8 7

14 69 40 15 9 13 11 29 19 21 1.4 3.4 3.5 2.5 10 8.5

15 64 39 14 8.

5

12 9 25 17 22 1.7 3.2 3.5 2.4 8 7.5


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THE GOLDEN RATIO 8

16 62 37 15 8 11 10 25 18 19 1.5 3.1 3.5 2.3 9 8

17 68 41 14 9 13 10.

5

27 19 22 1.3 3.4 3.6 2.4 9.5 8.5

18 65 40 14

.5

9 12

.5

10.

5

25 17 23 1.7 3.3 3.4

5

2.2 9.5 8

19 63 40 15 8 12 11 23 19 21 1.5 3.2 3.5 2.3 10 8.5

20 67 41 16 9 13 11 26 20 21 1.4 3.1 3.6 2.5 9.5 8

21 64 38 15 8.

5

12 10.

5

26 18 20 1.2 3.2 3.5 2.3 9.5 8.5

22 62 37 17 10 13 11 25 18 21 1.4 3 3.5 2.2 10 8

23 67 40 16 11 11 10 27 20 20 1.3 3.2 3.4 2.3 9 8

24 62 38 16 9 12

.5

11 24 20 18 1.5 3 3.6 2.2 10 9

25 66 40 16

.5

10 12 10.

5

26 19 21 1.7 3.3 3.5 2.4 9.5 8

26 65 39 15

.5

10 13 11 26 18 21 1.5 3.2 3.4

5

2.3 10 8.5

27 68 40 16 9 13

.5

11 28 17 23 1.5 3.2 3.5 2.4 10 9

28 63 40 15

.5

9.

5

11

.5

10 23 18 22 1.6 3.4 3.6 2.5 9 7.5

29 69 41 15 8 12

.5

10 28 20 21 1.3 3.3 3.4

5

2.2 9 8


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THE GOLDEN RATIO 9

30 68 41 14 8 13 10 27 18 23 1.2 3.2 3.5 2.5 9.5 8

Measurement Labels

A- Full Height

B- Naval-Foot

C- Finger Tip-Elbow

D- Wrist-Elbow

E- Shoulder-Top of Head

F- Head Length

G- Naval-Top of Head

H- Naval-Knee

I- Knee-End of Foot

J- First 2 Sections of Index Finger

K- Length of Index Finger

L- Length of Middle Finger

M- Length of Pinky

N- Length of Face

O- Width of Face

Figure 2: Summary Statistics Table for the Data Collected from the Measurements of 30

Participants. Each of the results follow the a+b/a ratio, however it was necessary to shorten the

label of each data in order to create a more legible table.


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THE GOLDEN RATIO 10

After using the Kruskal-Wallis technique to determine that there were significant differences

between my data sets I looked at the medians of each set. Since many of the medians were close

I could only conclude that several or most of them were different from each other, but could not

conclude that only one was significantly different from the rest.

Figure 3:

For each set of measurements said to correlate to the golden ratio of 1.618 I calculated the ratio

for each participant. None of the ratios match the golden ratio of 1.618.


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THE GOLDEN RATIO 11

Full Height + Naval-Foot / Full Height Finger Tip-Elbow+Wrist-Elbow Shoulder-head+Head Length

1.5616 1.6875 1.7692

1.65 1.6 1.9

1.6557 1.625 1.7727

1.5692 1.5625 1.6923

1.5652 1.5333 1.83331.629 1.6 1.9091

1.597 1.5882 1.9167

1.6029 1.5517 1.7308

1.6032 1.5806 1.9091

1.5692 1.5 1.8

1.5846 1.7143 1.8462

1.6143 1.6875 1.7407

1.6212 1.5625 1.8333

1.5797 1.6 1.8462

1.6094 1.6071 1.75

1.5968 1.5333 1.9091

1.6029 1.6429 1.8077

1.6154 1.6207 1.84

1.6349 1.5333 1.9167

1.6119 1.5625 1.8462

1.5938 1.5667 1.875

1.5968 1.5882 1.8462

1.597 1.6875 1.9091

1.6129 1.5625 1.88

1.6061 1.6061 1.875

1.6 1.6452 1.8462

1.5882 1.5625 1.8148

1.6349 1.6129 1.8696

1.5942 1.5333 1.8

1.6029 1.5714 1.7692


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Naval-Head+Shoulder-Head Naval-Knee+Knee-Foot 2sectionsofindex+index

1.4062 1.8636 1.5152

1.4762 1.95 1.4688

1.5238 1.7391 1.4516

1.4643 1.7619 1.4667

1.4 1.7727 1.4688

1.4783 1.8571 1.5667

1.4444 1.9048 1.5

1.4815 2.1579 1.4516

1.44 1.8095 1.4242

1.4464 1.85 1.4688

1.4815 1.9 1.4516

1.5 1.8696 1.4615

1.48 1.7826 1.4375

1.4483 1.9048 1.4118

1.48 1.7727 1.53121.44 1.9474 1.4839

1.4815 1.8636 1.3824

1.5 1.7391 1.5152

1.5217 1.9048 1.4688

1.5 1.9524 1.4516

1.4615 1.9 1.375

1.52 1.8571 1.4667

1.4074 2 1.4062

1.5208 2.1111 1.5

1.4615 1.9048 1.51521.5 1.8571 1.4688

1.4821 1.7391 1.4688

1.5 1.8182 1.4706

1.4464 1.9524 1.3939

1.4815 1.7826 1.375


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Figure 4:

As you can see by the histogram of the Full Height + Naval-Foot/ Full Height ratios there are 5

people whose measurement ratios fall between 1.6116 and 1.6216 which is close to the golden

ratio of 1.618. However since our research desires an actual ratio result of 1.618 to conclude that

there is a match to the golden ratio we can see by our ratio data that none of the ratios match

exactly.

Middle Finger+Pinky FaceLength+Width Naval-Foot+Naval-Top of Head

1.7429 1.7778 1.7805

1.7059 1.8 1.5385

1.6765 1.875 1.525

1.5882 1.8889 1.7568

1.6286 1.8333 1.7692

1.7143 1.8421 1.5897

1.7353 1.8 1.675

1.7077 1.875 1.6585

1.6857 1.8889 1.6579

1.7059 1.7778 1.7568

1.6286 1.85 1.7105

1.7143 1.8889 1.6279

1.6667 1.875 1.6098

1.7143 1.85 1.725

1.6857 1.9375 1.6411.6571 1.8889 1.6757

1.6667 1.8947 1.6585

1.6377 1.8421 1.625

1.6571 1.85 1.575

1.6944 1.8421 1.6341

1.6571 1.8947 1.6842

1.6286 1.8 1.6757

1.6765 1.8889 1.675

1.6111 1.9 1.6316

1.6857 1.8421 1.651.6667 1.85 1.6667

1.6857 1.9 1.7

1.6944 1.8333 1.575

1.6377 1.8889 1.6829

1.7143 1.8421 1.6585


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Figure 5: This histogram shows the ratio results of the Finger Tip-Elbow + Wrist-Elbow/Finger

Tip-Elbow data. There is only one person whose measurement ratio is close to the golden ratio of

1.618 which if we look at our ratio data list we can see that persons ratio is 1.6129 which is not

a match for our golden ratio.


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Figure 6: For the Shoulder Line-Top of Head + Head Length/ Shoulder Line-Top of Head

histogram we see that none of the participants measurements resulted in a ratio near 1.618.


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THE GOLDEN RATIO 17

Figure 7: For the Naval-Top of Head + Shoulder Line-Top of Head/Naval-Top of Head ratio

data none of the participants measurements had a ratio near 1.618.


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Figure 8: The Naval-Knee + Knee-End of Foot/Knee-End of foot ratio data shows that none of

the participants measurement ratios were near the golden ratio of 1.618.


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Figure 9: For the first 2 sections of the index finger + Length of the index finger/ Length of the

index finger ratio data none of the participants measurement ratios are near the golden ratio of

1.618.


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THE GOLDEN RATIO 20

Figure 10: For the Length of Face + Width of Face/ Length of Face ratio data none of the

participants measurement ratios were near the golden ratio of 1.618.


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THE GOLDEN RATIO 21

Figure 11: The Naval-Foot + Naval-Top of Head/ Naval-Foot ratio data shows that one

participants measurement ratio was between 1.6050 and 1.6250, but looking at our measurement

ratio chart the one persons ratio is 1.6098 which is not a match to the golden ratio of 1.618.

Figure 12: With these box and whisker plots of the measurement ratios of our data sets we can

see that the medians of each set of data are not similar showing that there is not a consistency of

any type of golden ratio between the sets of data.


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Appendix:

Our conclusion is based on the P-value of 0 calculated by our Kruskal-Wallis technique

compared to our alpha of 0.05 which tells us to reject our null hypothesis (HO: c= p= z) that

states that all of our data should be equal and accept our alternative hypothesis (HA: At least one

) that at least one of the data sets is different than the others.

Sources

Figure 1: Raw Data and Measurement LabelsFigure 2: Summary Statistics Table for collected data

Figure 3: Participants Measurement Ratio DataFigure 4: Full Height + Naval-Foot/ Full Height ratio data histogram

Figure 5: Finger Tip-Elbow + Wrist-Elbow/Finger Tip-Elbow data histogramFigure 6: Shoulder Line-Top of Head + Head Length/ Shoulder Line-Top of Head data histogram

Figure 7: Naval-Top of Head + Shoulder Line-Top of Head/Naval-Top of Head data histogramFigure 8: Naval-Knee + Knee-End of Foot/Knee-End of foot data histogram

Figure 9: First 2 sections of the index finger + Length of the index finger/ Length of the indexfinger data histogram

Figure 10: Length of Face + Width of Face/ Length of Face data histogramFigure 11: Naval-Foot + Naval-Top of Head/ Naval-Foot data histogram

Figure 12: Box-and-Whisker plots of the measurement ratio dataGolden Ratio Phase 1:http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1
http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1


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References

Harris, William. (n.d.). Leonardo of Pisa The Fibonacci Numbers. Retrieved from

http://community.middlebury.edu/~harris/fibonacci.html

Parveen, Nikhat. (n.d.) Golden Ratio Used by Greeks. Retrieved from

http://jwilson.coe.uga.edu/EMAT6680/Parveen/Greek_History.htm

The Golden Ratio. Retrieved from http://www.geom.uiuc.edu/~demo5337/s97b/art.htm
http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1http://katiev1985.wikispaces.com/Golden+Ratio+Phase+1

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