Upload
katie-votaw
View
218
Download
0
Embed Size (px)
Citation preview
7/31/2019 Golden Ratio Project Phase 2
1/23
Running head: THE GOLDEN RATIO 1
The Golden Ratio
Katie Votaw
Stats-1510
Taft College
7/31/2019 Golden Ratio Project Phase 2
2/23
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.
7/31/2019 Golden Ratio Project Phase 2
3/23
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
7/31/2019 Golden Ratio Project Phase 2
4/23
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,
7/31/2019 Golden Ratio Project Phase 2
5/23
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
7/31/2019 Golden Ratio Project Phase 2
6/23
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.
7/31/2019 Golden Ratio Project Phase 2
7/23
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
7/31/2019 Golden Ratio Project Phase 2
8/23
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
7/31/2019 Golden Ratio Project Phase 2
9/23
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.
7/31/2019 Golden Ratio Project Phase 2
10/23
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.
7/31/2019 Golden Ratio Project Phase 2
11/23
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
7/31/2019 Golden Ratio Project Phase 2
12/23
THE GOLDEN RATIO 12
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
7/31/2019 Golden Ratio Project Phase 2
13/23
THE GOLDEN RATIO 13
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
7/31/2019 Golden Ratio Project Phase 2
14/23
THE GOLDEN RATIO 14
7/31/2019 Golden Ratio Project Phase 2
15/23
THE GOLDEN RATIO 15
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.
7/31/2019 Golden Ratio Project Phase 2
16/23
THE GOLDEN RATIO 16
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.
7/31/2019 Golden Ratio Project Phase 2
17/23
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.
7/31/2019 Golden Ratio Project Phase 2
18/23
THE GOLDEN RATIO 18
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.
7/31/2019 Golden Ratio Project Phase 2
19/23
THE GOLDEN RATIO 19
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.
7/31/2019 Golden Ratio Project Phase 2
20/23
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.
7/31/2019 Golden Ratio Project Phase 2
21/23
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.
7/31/2019 Golden Ratio Project Phase 2
22/23
THE GOLDEN RATIO 22
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+17/31/2019 Golden Ratio Project Phase 2
23/23
THE GOLDEN RATIO 23
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