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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

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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

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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

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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

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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

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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


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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


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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 3.1.1 Figure 3.1.1

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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


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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



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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



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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.


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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



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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



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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


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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.


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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


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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



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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



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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


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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.

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