stevenhsu.weebly.comstevenhsu.weebly.com/.../9/26398796/golden_ratio.docx · web vieweven though...
Post on 30-Jan-2018
221 Views
Preview:
TRANSCRIPT
HSU
Access International Academy Ningbo
RESEARCH PAPER
SUBJECT: Mathematics
RESEARCH QUESTION : Does the Golden Ratio
influence us?
NAME: Steven Hsu
INSTRUCTOR: Ms. Jennie
DATE: January 2014
WORD COUNT: 3930
i
HSU
Abstract
A Golden Ratio, or Phi (φ), is a set of numbers that can extend for many pages. Usually it
is defined as, or rounded up to, 1.618. This ratio existed in many forms and in many places and
fields. It existed in history, in art, in nature and even in our lives. The Golden Ratio, however, is
often being recognized as just a ratio. Most people think it is just an ordinary ratio, likes other
ratios in mathematics, and do not realize the wide existence of the Golden Ratio.
If the Golden Ratio is widely displaced, then does the Golden Ratio have certain effects
on us? We, as humans, do not recognize the Golden Ratio consciously, however, we do
recognize it unconsciously. If we observed with care then an interesting phenomenon can be
discovered. The search of the Golden Ratio will take place in history, in nature and in daily life.
Furthermore, there will be precise examples and concise analysis in each field. Different forms
of Golden Ratio is discovered and used in a way that is not extremely concealed. In other words,
we are constantly and unconsciously expose to the Golden Ratio. Just a side note, the title and
the context of this page is in the Golden Ratio. The appearance of the Golden Ratio in all we see,
experience and create has unconsciously establishes a sense of harmony, balance, and beauty in
our life and nature.
Word count: 238
ii
HSU
Table of Contents
Introduction 1
Golden Ratio and Fibonacci Numbers 2
Golden Ratio in History 4
Golden Ratio in Nature 9
Golden Ratio in Daily Life 14
Conclusion 18
Bibliography 19
iii
HSU
1. Introduction
A Golden Ratio, or Phi (φ), is a set of numbers that can extend for many pages. Usually it is
defined as, or rounded up to, 1.618. With this unbounded irrational number, there is a question
that develops. “Does the Golden Ratio influence us?” This question is worth studying because
usually people think that the Golden Ratio is just a ratio, however, some other people think it
exists in our daily lives and constantly influences us. To determine the impact of the Golden
Ratio, I will first prove the existence of Golden Ratio in the field around us. I will find its
existence in art, in architectures, in nature, and in our society. After the existence is proven, then
the importance of impact can be easily concluded. If we are living in a world where people are
constantly exposed to the Golden Ratio, then we are likely to use the Golden Ratio
unconsciously in our daily life. “The CN Tower in Toronto…has [incorporated] the golden ratio
in its design. The ratio of observation deck at 342 meters to the total height of 553.33 is 0.618 or
phi” (Owen). See figure 1.1 for the actual picture for the CN Tower.
2. Golden Ratio and Fibonacci Numbers Figure 1.1Figure 1.1Figure 1.1
1
HSU
2.1. Golden Ratio:
The Golden Ratio also known as Golden Proportion, Golden Selection, Golden Mean and Divine
Proportion. The Golden Ratio, represented as Phi (φ), is a ratio that is round up to 1.618. Phi is a
ratio that continues forever and without repeating; which called irrational number. This ratio can
be found through different ways.
One of the most symbolic
ways is through the ratio of the
length of a segment. “Golden
Ratio…results when a line is
divided in one very special and unique way” (Meisner). As shown in Figure 2.1.1, when
segment A is separated into segment B and C in a particular way, the
Golden Ratio is created. Sometime 0.618 and 0.382 can also be
recognized as Golden Ratio. This is because they are components of
forming actual Golden Ratio or Phi.
Besides from segments, Golden Ratio can also be generated
through the concept of Golden Rectangle. In Figure 2.1.2, the concept of
Golden Rectangle and Golden Ratio can be visualized. If a square is cut
out from the Golden Rectangle, then another Golden Rectangle is
formed. This procedure can be repeated and received same result. As
the square is removed, the ratio of the square and the new Golden
Rectangle is Phi.
Another form of the Golden Ratio is the Golden Angles.
As shown in Figure 2.1.3, a circle is being divided into two
sections. The 222.5 section is also known as 0.618 turns of a
Figure 2.1.1
Figure 2.1.2Figure 2.1.2Figure 2.1.2
Figure 2.1.1Figure 2.1.1
2
HSU
circle and the 137.5 section is also known as 0.382 turns of a circle. Both sections can be
considered as Golden Angle because the Golden Ratio can be formed when the ratio of these two
sections is formulated.
2.2. Fibonacci Numbers:
The Fibonacci Numbers is, sometime known as Fibonacci Series and Fibonacci Sequence, “A
sequence of numbers in which each number is the sum of the two preceding numbers, e.g.
0,1,1,2,3,5,8,…”( Daintith). In other words, the next number in the series can be found by adding
the two previous numbers before it or through the equation of F0=0,F1=1,Fn=F(n-1)+F(n-2);
(n>=2).
Another way to find the sequence is through Pascal’s Triangle. As shown in Figure 2.2.1,
Fibonacci Numbers can be obtained by adding the diagonal numbers.
2.3. Their
Relationship:
The Golden Ratio and Fibonacci Numbers are closely related. Even though they looked different
and have usage, but they can end up with same result. “The [G]olden [S]ection number is closely
Figure 2.2.1Figure 2.2.1Figure 2.2.1
3
HSU
connected with the Fibonacci series and has a value of (5 + 1)/2” (Knott). In other words, the
Golden Ratio is ± 1.618 and Fibonacci Numbers can also result 1.618. The way Fibonacci
Numbers result in 1.618 is through the average ratio between successive Fibonacci Numbers.
Another way to show their relationship is through rebuilding Golden Rectangle. As
shown in Figure 2.3.1, using Fibonacci Numbers allow us to reconstruct the Golden Rectangle.
The reason Golden Rectangle can be formed by Fibonacci Numbers is because they have same
ratio. In Figure 2.3.1, the concept of reconstructing Golden Rectangle can be visualized.
Moreover, if the sequence in Figure 2.3.1 is reversed, the concept of Golden Rectangle reducing
can be visualized; it will be the same concept as shown in Figure 2.3 and Figure 2.1.2. “Spiral
shells also exhibit patterns related to the Fibonacci sequence” (Smoller). This concept of
reconstructing Golden Rectangle with Fibonacci Numbers can also be known as the formation of
Golden Spiral. The concept and the formation of Golden Spiral are shown in Figure 2.3.2.
3. Golden Ratio in History
Golden Ratio is not something just discovered recently. It actually existed in various famous
artworks and architectures in our history.
3.1 In Famous Artworks
Figure 2.3.2Figure 2.3.1
4
HSU
One of the earliest forms of Golden Ratio existed in famous artworks. The existences of Golden
Ratio can be observed and discovered in different famous artworks.
One of the famous artwork that contains Golden Ratio is
Leonardo da Vinci (1452-1519)’s Mona Lisa. The Golden
Ratio in Mona Lisa existed in the form of Golden Rectangle.
As shown in Figure 3.1.1, Mona Lisa included Golden
Rectangles. In addition, this Golden Rectangles created
significant purpose for the painting. “… [T]he edges of these
new squares come to all the important focal points of the
woman: her chin, her eye, her nose, and the upturned corner of
her mysterious mouth…” ("The Fibonacci Series."). The Focal
points help to bring attentions to particular part of the painting.
However, Mona Lisa included more than one set of Golden
Rectangles. Another set of Golden Rectangles can be
discovered from the close view of the woman’s face.
As shown in Figure 3.1.2, the woman’s face can be
divided into many Golden Rectangles. Similarly as the
purpose in Figure 3.1.1, the Golden Rectangles
existed in the face created numerous amounts of focal
points. “[These focal points helped] to create a sense
of beauty and balance…” ("Could You Explain the
Most Basic Types of Balance Used in
Compositions?").
Figure 3.1.1
Figure 3.1.2Figure 3.1.2Figure 3.1.2
Figure 3.1.1 Figure 3.1.1
5
HSU
Besides from Leonardo da Vinci’s Mona Lisa,
Golden Rectangles also existed in other artists’ artworks.
Joseph Mallord William Turner (1775-1851)’s Norham
Castle at Sunrise also contained Golden Rectangle. At the
first glimpse, this artwork may not indicate any sign or clue
of Golden Rectangle. However, the Golden Rectangle can
be discovered as the viewer’s attentions gradually draw to the brownish creature. As shown in
Figure 3.1.3, the brownish creature actually marks the borderline of two Golden Rectangles.
“Joseph Mallord William Turner is admired for his use of color and light… [and these] particular
interests are the geometric similarities in his various canvases…” (Britton). In other words, even
though Joseph Mallord William Turner is well-known
for “his use of color and light,” but his usage of Golden
Rectangle do existed in most of his artworks and have
significant influences.
French neo-impressionist Seurat’s (1859-1891)
“Bathers” is another famous artwork that contain a
Golden Rectangle. As shown in Figure 3.1.4, the Bathers
includes many Golden Rectangles. The three main
people in the painting are formed in the way of Golden
Rectangles. From the head to the waist, each person created an unconscious Golden Rectangle.
Moreover, the whole painting can be divided into four Golden Rectangles. The boundaries are
set by the horizon and the head of the person in the middle.
Golden Rectangles exist in different famous paintings that we normally think of as
beautiful. The importance of Golden Rectangle can be observed in various famous artworks and
Figure 3.1.3Figure 3.1.3Figure 3.1.3
6
HSU
it has dramatic influence on the paintings. Perhaps, just like Luca Pacioli said, “without
mathematics there is no art” (Meisner).
3.2 In Ancient Architectures
Besides from famous artworks, Golden Ratio can also be discovered in numerous amounts of
ancient architectures.
The most classic example is the
Parthenon in Acropolis, Athens. The
Parthenon has different form of Golden
Ratio existed in different section or
part of the Parthenon. The first
existence of Golden Ratio is at its main
entrance. Moreover, its main entrance
can show two forms of Golden Ratio. As shown in Figure 3.2.1, the Golden Ratio discovered in
the main entrance of Parthenon are
Golden Rectangle and Golden Spiral.
Another part of Parthenon that
contains Golden Ratio is the columns.
The columns of Parthenon have Golden
Ratio in the form of Golden Rectangles.
“…The width of the columns is in a
golden ratio proportion formed by the
distance from the center line of the columns to the outside of the columns…” (Meisner). This
existence can be visualized in Figure 3.2.2.
Figure 3.2.1
Figure 3.2.2
Figure 3.2.1Figure 3.2.1
Figure 3.2.2Figure 3.2.2
7
HSU
If the top of the columns is magnified, then another
example of Golden Ratio will appeared. The top of column
can be divided into three arrangements of Golden Ratio.
The arrangements are created by the design and the
boundary of the Parthenon. One is formed by the section that
is above the columns, shown in Figure 3.2.3 as vertical
rectangle. Another one is formed by the carving that is
between the boundaries, shown in Figure 3.2.3 as horizontal
rectangle.
Another arrangement of Golden Ratio in this section is shown in Figure 3.2.4. The Golden Ratio
existed in the form of Golden Spiral and it is similar to the horizontal rectangle shown in Figure
3.2.3.
The Golden ratio in the Parthenon did
not just exist in the outer form. In other
words, the Parthenon includes some Golden
Ratios that cannot be discovered from outside
views. The way Parthenon is arranged contain
many Golden Ratios; in the form of both
Golden Rectangles and Golden Spirals. In
Figure 3.2.5, a floor plan of the Parthenon is shown. From the floor plan, many Golden
Rectangles and Golden Spirals can be identified.
The Parthenon contain dramatic amounts of the Golden Ratio. These dramatic existences
may seem as the architects designed it on purpose. However, this conjecture is invalid. “If… the
golden ratio was intended to be included among the many numbers and proportions included,
Figure 3.2.4
Figure 3.2.5
Figure 3.2.4Figure 3.2.4
Figure 3.2.5Figure 3.2.5
8
HSU
then one can find some rather compelling evidence that they applied it… with the deeper
knowledge recorded by Euclid…150 years later” (Meisner).
4. Golden Ratio in Nature
Golden Ratio also existed in the field of Nature. The existence of Golden Ratio in Nature is
created naturally and constantly expose to human beings. The reason of existence is unknown
and remains uncanny. However, this ratio existed in an extremely conceal way; if the
observations are not carried carefully and precisely, then its existence cannot be discovered
easily.
4.1 In Plants:
The most obvious example among all the plant examples is the flower petals. The flower petals
include Fibonacci Numbers and Golden Ratio. Many species of flower have the same number of
petals as one of the numbers in the Fibonacci Numbers. Some famous examples are clovers-
even though they are not flowers and they have leaves instead of petals, but they correspond with
the Fibonacci Numbers- buttercups, chicory, and daisy; the clovers have three petals, buttercups
have five petals, chicories have 21 petals, and daisies have 34 petals (Dvorsky). As shown in
Figure 4.1.1 and mentioned above, petals, and leaves, do correspond with the Fibonacci
Numbers.
Figure 4.1.1Figure 4.1.1Figure 4.1.1
9
HSU
More importantly, Golden Ratio also appeared in the petals; it appeared in the form of
Golden Angle. Even though it required very precise and chary analysis, but we still discovered
its existences. “[Golden Ratio] appears in petals on account of …each petal is placed at 0.618034
per turn (out of a 360° circle)” (Dvorsky).
Besides from flowers, trees also have
Fibonacci Numbers in them. The Fibonacci
Numbers can be seen in the formation or the
arrangement of the tree branches. “This
pattern of branching is repeated for each of
the new stems. A good example is the
sneezewort. [The] [r]oot systems and [the]
algae exhibit this [kind of] pattern” (Dvorsky). As shown in Figure 4.1.2, the way trees are
branching follow the pattern of the Fibonacci Numbers. This pattern is extremely mysterious.
What is the chance of branching pattern follows the pattern of the Fibonacci Numbers? Is it mere
coincidence or something deeper?
4.2 In Animals:
The Golden Ratio and Fibonacci Numbers also existed in animals.
Among all the examples of Golden Ratio and Fibonacci Numbers in
animals, the most miraculous example is the animal body. The
Golden Ratio existed in animals in the form of proportion.
The Golden Ratio of a penguin is the most obvious example. The proportions existed in a
penguin is marked by the key body parts of penguins. Each section can be clearly separated by
the body features of penguin and formed the Golden Ratio. “The eyes, beak, wing and key body
Figure 4.1.2
Figure 4.2.1Figure 4.2.1Figure 4.2.1
Figure 4.1.2Figure 4.1.2
10
HSU
markings of [a] penguin all fall at golden sections [proportion to] its height” (Meisner). As
shown in Figure 4.2.1, a penguin can be divided into six sections that are significant examples of
Golden Ratio.
These kind of patterns also existed in other animals, such as, moths and ants. The pattern
on a particular kind of moth has marked the boundaries of
the Golden Ratio. The boundaries created by this particular
kind of moth have created the Golden Ratio of its width and
length. The sections are formed by the eye-like pattern on the moth and this phenomenon can be
visualized in Figure 4.2.2.
In comparison, ants have more Golden Ratio in them. Ants can have two Golden Ratios
in them. “The body sections of an ant are defined by the
golden sections of its length. [In addition,] [i]ts leg
sections are also golden sections of its length”
(Meisner). In other words, the first Golden Ratio can be
spotted through the body sections, similar to the Golden Ratio in a penguin. The second Golden
Ratio can be spotted through the leg sections; this Golden Ratio is relatively rare, since most
Golden Ratios in animals are in the way or form of body section or pattern sections. This rare
phenomenon can be seen in Figure 4.2.3.
Other than animal bodies or anatomies, Golden Ratio can also be discovered in animals’
reproductive dynamics. Two most famous examples are the idea rabbit reproductive pattern,
representation of Fibonacci Numbers, and the gender ratio of honey bees, representation of
Golden Ratio.
The idea rabbit reproductive pattern is actually a mathematic puzzle about hypothetical
rabbits. Even though this example is a hypothetical one, but it follow the Fibonacci Numbers and
Figure 4.2.2
Figure 4.2.3
Figure 4.2.2Figure 4.2.2
Figure 4.2.3Figure 4.2.3
11
HSU
mimic the exponential growth of a population. The question states “A certain man put a pair of
rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced
from that pair in a year if it is supposed that every month each pair begets a new pair which from
the second month on becomes productive?” (Smoller). The solution and Fibonacci Numbers can
be seen in the Figure 4.2.4.
The honey bees reproduce in an interesting way. The gender ratio of honey bees is a
representative example of Golden Ratio in reproductive dynamic. This example is not
hypothetical. “[Ratio of] the number of females in a colony by the number of males (females always
outnumber males)… is typically something very close to 1.618” (Dvorsky).
4.3 Other Natural Phenomena:
Nature includes more than animals and plants. This aspect is same with the Golden Ratio. The
Golden Ratios existed beyond animals and plants; the Golden Ratio also exited in non-living
things in the Nature, in the natural disasters, and in even outer spaces.
The most famous example among non-living things that
contain Golden Ratio is the shell. The Golden Ratio existed in shell
is in the form of Golden Rectangle and Golden Spiral. As shown in
Figure 4.3.1, a typical shell can have the existence of both Golden
Rectangle and Golden Spiral. The width and length of a shell can
form a Golden Rectangle easily and the curving boundaries of the
shell can form a Golden Spiral. Similar phenomenon can also be
discovered in the natural disasters.
In disasters, the most significant example is hurricane or the typhoon. In hurricane and
typhoon both Golden Rectangle and Golden Spiral can
Figure 4.3.1Figure 4.3.1Figure 4.3.1
12
HSU
be witnessed in a single existence. Even though hurricane and typhoon are dangerous and
disastrous, but they contain Golden Rectangle and Golden Spiral in them. As shown in Figure
4.3.2, the hurricane looks a typical shell in the Nature and includes both types of Golden Ratio.
A more surprising fact of the Golden Ratio is that it existed
beyond the Earth. In other words, it also can be seen in the
Universe. Many examples of Golden Ratio can be found in the
Universe, but the clear ones are in the Milky Way or the
Galaxy. “[S]piral galaxies also follow the familiar Fibonacci
[Numbers and the Golden Ratios]” (Dvorsky) Even though the
shape of a spiral galaxy looks like an ordinary shell, but it is a
significant example of the Golden Ratio that is beyond the
Earth.
5. Golden Ratio in Daily Life
Not surprisingly, the Golden Ratio is also hidden in our daily life. It has existed in our life for a
long time and not everyone knows the existence of Golden Ratio in daily routine. The Golden
Ratio, however, surprisingly existed in many famous products and many interesting phenomena
in finances and music.
5.1 In Finance:
Even though there is no direct example of Golden Ratio in finance phenomena, but Fibonacci
Numbers do existed in the finance field. One of the most famous aberrations in the finance field
is the Fibonacci Retracement.
Figure 4.3.3Figure 4.3.3Figure 4.3.3
13
HSU
The Fibonacci Retracement is a method of analysis in finance. The Fibonacci
Retracement is derived from Fibonacci Numbers. “Fibonacci numbers are important to
[investors] because [they] take note of the key ratios 0.382, 0.50, 0.618, 0.786, 1.00, 1.27, 1.618
[and] 2.618. They expect retracements to find support when the price drops 38.2%, 50%, 61.8%,
78.6%, 100%, 127%, 161.8% [and] 261.8%. Similarly, when a stock price has dropped it may
retrace to an extent related to these Fibonacci ratios” ("FIFTI™ Education Using Fibonacci."). In
other words, Fibonacci Retracement is widely used in finance and has relatively high reputation.
As mentioned above, the Fibonacci Retracement is used in finance and is quite reliable.
However, this method may sounds too ideal to be true for some experts. Even though there are
experts who do not believe in this method, but the Fibonacci Retracement does existed in reality.
The Figure 5.1.1 shows the price
movement of USD and CAD
currency pair between September
23, 2013 and September 24,
2013. During this time period, the
price retraced approximately
38.2%. As shown in Figure 5.1.1,
the Fibonacci Retracement does
exist.
Even though finance only
contain examples of Fibonacci Numbers, but the Golden ratio does exist in a tool that many
people use; the credit card. The credit card has a very good proportion of the Golden Ratio. As
the author of this research paper measures a typical credit card, he finds that the length is
approximately 86 millimeters and width is approximately 54 millimeters. When the ratio of
Figure 5.1.1Figure 5.1.1Figure 5.1.1
14
HSU
length and width is taken, the ratio is around 1.6. The ratio means that a typical credit card does
not just have credits in it, but also the Golden Ratio.
5.2 In Music:
This may sounds ridiculous, but Golden Ratios also exists in music! The existence of Golden
Ratio can be discovered in two main fields; the instruments
and the tools.
The representative example of Golden Ratio in the
instruments is the violin. In comparison, the existence of
Golden Ratio in violin is the simplest and most
understandable. As shown in Figure 5.2.1, the violin can be
divided into two sections and the ratio of the two
sections is Golden Ratio. As the author of this
research paper measures his own violin, he finds
out that the longer section is approximately 36
centimeters and the shorter section is
approximately 22.5 centimeters. Then the ratio
of the violin is around 1.6. Even though the ratio
is not a perfect Golden Ratio, but it is close
enough to call it a Golden Ratio.
Other than instruments, the Golden Ratio
also existed in musical related tools. One of the
most famous examples is the Cardas Audio.
“George Cardas founded the [Cardas Audio] to
Figure 5.2.1
Figure 5.2.2
Figure 5.2.1Figure 5.2.1
Figure 5.2.2Figure 5.2.2
15
HSU
perfect audio cables using ultra-pure materials, innovative Golden ratio resonance control
techniques and uniquely insightful solutions to transmission line problems” ( Cardas). In other
words, the Cardas Audio used the concept of Golden Ratio to improve the speakers or the audios.
Moreover, the Casrdas Audio does not only follow the
Golden Ratio when the company constructs the speakers, but
also provides setup plans that are based on the concept of
Golden Ratio. One of the official setup plans can be seen in
Figure 5.2.2.
The Golden Ratio existed in the speakers also helped the
company to win an award, as shown in Figure 5.2.3. Perhaps, the Golden Ratio is not just a ratio
but something deeper.
5.3 In Famous Modern Designs:
Among all the Golden Ratio examples, the existences of the Golden Ratio in famous modern
designs are the aberrations. Many famous modern designs do have Golden Ratio in them and
more importantly, they contained relatively more Golden Ratio examples then other fields.
The most famous example of
the Golden Ratio in modern designs is
the logo of the Apple Inc. “[The] Apple
[Inc.] ha[s] used the [Golden Ratio] in
designing their Logo” (Kditz, Malte).
The Apple Inc.’s logo contains
numerous amounts of Golden Ratio in
Figure 5.2.3
Figure 5.3.1
Figure 5.2.3Figure 5.2.3
Figure 5.3.1Figure 5.3.1
16
HSU
it; in form of Golden Rectangle, Golden Ratio and Golden Spiral. This fabulous phenomenon can
be visualized in Figure 5.3.1.
In Figure 5.3.1, the right part shows the actual logo of the Apple Inc. and the left part
shows the proportion of the curves used in the logo. In other words, the curves used in the logo
are proportioned to the sections in the Golden Spiral and corresponded to the numbers in the
Fibonacci Numbers.
Besides the tremendous numbers of Golden Ratios in Apple Inc.’s logo, the Golden Ratio
also existed in the iCloud logo. “[T]he new iCloud logo is heavily based on the Golden Ratio”
(Kditz, Malte). In comparison, the iCloud logo has less Golden Ratios than the Apple Inc. logo.
As shown in Figure 5.3.2, the iClound logo only has two main sets of Golden Ratio.
6. Conclusion
The Golden Ratio is not just a ratio; it is something deeper and more mysterious. The Golden
Ratio appears in history, in nature and in our daily life. The wide dissemination of the Golden
Ratio in many fields may not be just a coincidence, but something that exists intentionally. Is it
the signature of Natural forces? Or it is the result of something more superior; like God? The
reason of the formation of Golden Ratio does not matter. Most importantly the Golden Ratio has
existed on the Earth for a long time and the Golden Ratio does influence the human population
unconsciously. As a human, we tend to use and prefer the existence of Golden Ratio in our
Figure 5.3.2Figure 5.3.2Figure 5.3.2
17
HSU
creations and environments; it is something we had for an extremely long period of time. The
appearance of the Golden Ratio in all we see, experience and create has unconsciously
established a sense of harmony, balance, and beauty in our life and nature.
Bibliography
ARSA. "Lady Blunt, a Rare Stradivarius Violin Sets $15.9 Million Auction Record to Help
Japan Quake Relief." EXtravaganzi. N.p., 21 June 2011. Web. 18 Dec. 2013.
<http://www.extravaganzi.com/lady-blunt-a-rare-stradivarius-violin-sets-15-9-million-
auction-record-to-help-japan-quake-relief/>.
Britton, Jill. "TITLE." N.p., 6 May 2012. Web. 18 Dec. 2013.
<http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm>.
Cardas, George. "Cardas Audio." Cardas Audio. N.p., n.d. Web. 18 Dec. 2013.
<http://www.cardas.com/welcome.php>.
Daintith John. "Fibonacci series." A Dictionary of Computing. 2004. Encyclopedia.com. 8 Dec.
2013 <http://www.encyclopedia.com/topic/Fibonacci_series.aspx#1>.
Dvorsky, George. "15 Uncanny Examples of the Golden Ratio in Nature." Io9.com. N.p., 20 Feb.
2013. Web. 18 Dec. 2013. <http://io9.com/5985588/15-uncanny-examples-of-the-golden-
ratio-in-nature>.
Jordan, Michele Anna. "Your St. Patrick’s Day Traditions + My Recipes." Eat This Now. N.p.,
15 Mar. 2012. Web. 18 Dec. 2013. <http://pantry.blogs.pressdemocrat.com/13172/your-
st-patricks-day-traditions-my-recipes/>.
Kditz, Malte. "Golden Section in the Apple." Malte Kditz. N.p., 22 July 2011. Web. 18 Dec.
2013. <http://www.maltekoeditz.com/index.php/2011/07/22/golden-section-in-the-
apple/>.
18
HSU
Knott, R. "Fibonacci Numbers and the Golden Section." N.p., n.d. Web. 8 Dec. 2013.
<http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html>.
Meisner, Gary. "Quotes Related to Phi." Phi 1618 The Golden Number. N.p., 13 May 2012.
Web.
18 Dec. 2013. <http://www.goldennumber.net/phi-quotations/>.
Meisner, Gary. "The Golden Ratio An Overview of Its Properties, Appearances and
Applications." Phi 1618 The Golden Number. N.p., 13 May 2012. Web. 4 Dec. 2013.
<http://www.goldennumber.net/golden-ratio/>.
Meisner, Gary. "The Golden Section in Nature: Animals." Phi 1618 The Golden Number. N.p.,
13 May 2012. Web. 18 Dec. 2013. <http://www.goldennumber.net/nature/>.
Meisner, Gary. "The Parthenon and Phi, the Golden Ratio." Phi 1618 The Golden Number. N.p.,
20 Jan. 2013. Web. 18 Dec. 2013. <http://www.goldennumber.net/parthenon-phi-golden-
ratio/>.
Owen, John. "Phi and the Golden Section in Architecture." Phi 1618 The Golden Number. N.p.,
5 Aug. 2013. Web. 27 Nov. 2013. <http://www.goldennumber.net/architecture/>.
Peter. "Giant White Daisy Bloom." Cactus Blog. N.p., 30 Sept. 2008. Web. 18 Dec. 2013.
<http://www.cactusjungle.com/blog/2008/09/>.
Scharman, Jim. "Fibonacci Retracements." Profits Run Learn Stock Trading Forex Trading
Online FX Signals RSS. N.p., 23 Sept. 2013. Web. 18 Dec. 2013.
<http://www.profitsrun.com/featured/fibonacci-retracements/>.
Smoller, Laura. "The Fibonacci Sequence and the Golden Mean." The Fibonacci Sequence and
the Golden Mean. University of Arkansas at Little Rock, June 2001. Web. 18 Dec. 2013.
<http://ualr.edu/lasmoller/fibonacci.html>.
"Could You Explain the Most Basic Types of Balance Used in Compositions?" Saylor.org. N.p.,
19
HSU
n.d. Web. 18 Dec. 2013. <http://www.saylor.org/site/wp-
content/themes/saylor/curriculum/curriculumQAAJAX.php?action=getcourseunitqas>.
"FIFTI™ Education Using Fibonacci." Investing in the Stock Market with Investors Internet
Inc. N.p., n.d. Web. 18 Dec. 2013. <http://www.investorsinternet.com/fifti-
education/usingfibonacci.html>.
"The Fibonacci Series." ThinkQuest. Oracle Foundation, n.d. Web. 18 Dec. 2013.
<http://library.thinkquest.org/27890/applications6.html>.
20
top related