Places You Won't Believe The Golden Ratio & Fibonacci Sequence Appears

Summary

The video script delves into the omnipresence of the Golden Ratio, 1.618, in various facets of life, from the architectural marvels of the Great Pyramids and Taj Mahal to the artistic genius of Leonardo da Vinci. It explores its presence in nature, from spiral galaxies to the arrangement of petals in flowers, and its influence on human aesthetics and facial proportions. The script also highlights the Golden Ratio's role in music, advertising, and even DNA, showcasing its universal significance and its integration into human creativity and the natural world.

Takeaways

• 📏 The Golden Ratio, approximately 1.618, is a mathematical proportion found extensively in nature, art, architecture, and even in human biology.
• 🏛 The Great Pyramids of Giza exhibit the Golden Ratio in their geometry and placement, suggesting a deliberate use of this mathematical concept in ancient Egyptian architecture.
• 🏰 The Taj Mahal incorporates the Golden Ratio in its design, reflecting the significant contribution of Indian mathematicians to the field.
• 🖼 The Mona Lisa by Leonardo da Vinci is a renowned example of the Golden Ratio's use in art, with the painting's composition and the subject's facial structure adhering to this proportion.
• 🌌 The Golden Ratio appears in the cosmos, with the spiral shapes of galaxies, including our Milky Way, following a logarithmic growth pattern based on the Golden Ratio.
• 🌼 In nature, the Fibonacci Sequence and the Golden Ratio are evident in the arrangement of flower petals and the spiral patterns of plants like pinecones and pineapples.
• 🧬 The Golden Ratio is present within human DNA, with the dimensions of DNA molecules and the structure of the DNA helix reflecting this mathematical ratio.
• 🎵 Music is another domain where the Golden Ratio is believed to enhance the aesthetic appeal, with composers like Beethoven and Mozart reportedly using it in their compositions.
• 🤩 The human face's attractiveness is often associated with the Golden Ratio, with facial symmetry and proportions that adhere to this ratio being perceived as more appealing.
• 📐 The use of the Golden Ratio in advertising and design is prevalent, with companies leveraging its aesthetic appeal to create more engaging logos, layouts, and advertisements.
• 🦋 The anatomy of living things, including humans and various animals, often reflects the Golden Ratio, indicating a universal pattern in biological structures.

Q & A

What is the Golden Ratio and what is its value?

The Golden Ratio, often denoted by the Greek letter Phi (φ), is a mathematical proportion that is approximately 1.618. It is found in various aspects of art, architecture, and nature, and is based on the Fibonacci Sequence.

What is the Fibonacci Sequence and how does it relate to the Golden Ratio?

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1. It is closely related to the Golden Ratio because as the sequence progresses, the ratio of consecutive Fibonacci numbers approximates the Golden Ratio.

How is the Golden Ratio applied in the architecture of the Great Pyramids of Giza?

The Golden Ratio is seen in the geometry of the pyramids, with the height to base ratio of the internal triangles being close to the Golden Ratio. Additionally, the placement of the pyramids follows a phi matrix grid, indicating a deliberate use of the Golden Ratio in their orientation.

What is the significance of the Golden Ratio in the Taj Mahal?

The Taj Mahal incorporates the Golden Ratio in its architecture, with its rectangular base and other buildings on the premises following the phi matrix. This use of the Golden Ratio contributes to the aesthetic appeal and harmony of the structure.

How does Leonardo da Vinci utilize the Golden Ratio in his artwork, particularly in the Mona Lisa?

Leonardo da Vinci used the Golden Ratio in the composition of the Mona Lisa, with the painting's dimensions and the positioning of the subject adhering to the Golden Rectangle. Additionally, a Golden Spiral can be drawn from the center of her nose over her head and through her shoulders.

In what way does the Golden Ratio manifest in spiral galaxies?

The Golden Ratio appears in the form of the Golden Spiral, which is observed in the structure of spiral galaxies, including our own Milky Way. The spiral arms of these galaxies exhibit a logarithmic growth by a factor of phi as they expand.

Why are the Fibonacci Numbers and the Golden Ratio found in the arrangement of flower petals?

Many flowers exhibit the Fibonacci Numbers in the number of their petals or seeds, and the Golden Spiral can be seen in the overall shape of the flower. This arrangement is believed to optimize the exposure to sunlight and resources for the plant.

How is the Golden Ratio present in human DNA?

The Golden Ratio is found in the dimensions of human DNA, which measures 34 angstrom units long by 21 angstrom units wide, both Fibonacci Numbers. Additionally, the cross-sectional view of a DNA molecule shows a decagonal shape, which is related to the Golden Ratio.

How does the Golden Ratio influence music composition?

The Golden Ratio is used in the timing and pacing of some of the world's most popular music pieces, with the belief that compositions adhering to a Golden Ratio pattern are more pleasing to the ear. Examples include works by Beethoven and Mozart.

What role does the Golden Ratio play in human facial attractiveness?

The Golden Ratio is believed to contribute to facial symmetry, which is often associated with attractiveness. The proportions of an attractive face often fit within the phi matrix, and this principle is used by plastic surgeons and in facial recognition software.

How is the Golden Ratio used in advertising and design?

The Golden Ratio is employed in advertising and design to create visually appealing layouts and compositions. It can be found in logos of popular brands and is a fundamental aspect of web and graphic design, often used in template structures.

What is the connection between the Golden Ratio and the anatomy of living things?

The Golden Ratio is observed in the proportions of the human body and other living organisms. For example, the Vitruvian Man by Leonardo da Vinci illustrates the human body's adherence to the Golden Ratio, and similar proportions can be seen in the segmented bodies of insects.

Outlines

00:00

🔍 The Golden Ratio in Architecture and Art

The script discusses the prevalence of the Golden Ratio, approximately 1.618, in various aspects of life, starting with its appearance in the Great Pyramids of Giza. It suggests that the pyramids' geometry and orientation are based on the Golden Ratio, as seen in the near-perfect Kepler Triangle and the phi matrix grid. The script also highlights the Taj Mahal's use of the Golden Ratio in its architecture, emphasizing the cultural significance of mathematics in Indian history. Additionally, the Mona Lisa by Leonardo da Vinci is noted for its use of the Golden Ratio, with the painting's composition and the artist's other works reflecting mathematical harmony.

05:00

🌌 The Golden Ratio in Nature and the Cosmos

This section of the script explores the Golden Ratio's presence in the natural world and the cosmos. It mentions the logarithmic growth pattern of spiral galaxies, including our Milky Way, and how the shapes of the Earth and Moon relate to the Golden Ratio. The script also describes the Golden Spiral seen in hurricanes from space and the Fibonacci Series' occurrence in the arrangement of flower petals and seeds. Pine cones and pineapples are highlighted for their Fibonacci spiral patterns, and the DNA's structure is noted for its Golden Ratio components, suggesting a fundamental role in life's design.

10:02

🎵 The Golden Ratio in Music, Human Anatomy, and Design

The script delves into the Golden Ratio's influence on music, suggesting that its use in timing and pacing contributes to the pleasing nature of certain compositions, with examples from Beethoven and Mozart. It also touches on the Golden Ratio's role in human facial attractiveness and the use of facial symmetry in advertising and design. The script further discusses the Golden Ratio in the anatomy of living things, referencing Leonardo da Vinci's Vitruvian Man and the proportionality of the human body. It concludes with a mention of the Golden Ratio in animal body proportions and an invitation for viewers to share their insights on the topic.

Keywords

💡Golden Rule

The 'Golden Rule' is a moral guideline that encourages treating others as one would like to be treated. In the context of the video, it is used as an introduction to the 'Golden Ratio,' which is the main focus of the video. The script contrasts the well-known ethical principle with the lesser-known mathematical concept that plays a significant role in various aspects of life.

💡Golden Ratio

The 'Golden Ratio,' often represented by the number 1.618, is a mathematical proportion that appears in various natural and man-made structures. It is derived from the Fibonacci Sequence and is believed to be aesthetically pleasing. The video discusses its prevalence in architecture, art, and nature, emphasizing its importance in the design of the Great Pyramids of Giza and the Taj Mahal, among other examples.

💡Fibonacci Sequence

The 'Fibonacci Sequence' is a series of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1. It is closely related to the Golden Ratio, as the ratio of consecutive Fibonacci numbers approximates the Golden Ratio as the sequence progresses. The script uses the sequence to explain the mathematical basis for the Golden Ratio's ubiquity in nature and art.

💡Great Pyramids of Giza

The 'Great Pyramids of Giza' are ancient Egyptian pyramids and a prominent symbol of Egypt's cultural heritage. The video script suggests that the pyramids' construction and orientation incorporate the Golden Ratio, indicating a deliberate use of this mathematical principle in their design, which may have been intended for aesthetic or symbolic purposes.

💡Taj Mahal

The 'Taj Mahal' is a famous Indian mausoleum known for its architectural beauty and symmetry. The script mentions that the Taj Mahal's architecture successfully implements the Golden Ratio, particularly in its rectangular base and the orientation of its buildings, highlighting the integration of mathematics and art in its design.

💡Mona Lisa

The 'Mona Lisa' is a renowned painting by Leonardo da Vinci, which the script notes as an example of the Golden Ratio's use in art. The painting's composition and the subject's facial features are said to align with the Golden Ratio, contributing to its aesthetic appeal and the painting's iconic status.

💡Spiral Galaxies

The term 'Spiral Galaxies' refers to galaxies that have a spiral structure, such as the Milky Way. The video script points out that the Golden Spiral, a logarithmic spiral that grows by a factor of the Golden Ratio, can be observed in the structure of these galaxies, suggesting a universal pattern in celestial bodies.

💡Flower Petals

The number of 'Flower Petals' on many plants follows the Fibonacci Sequence, which is directly related to the Golden Ratio. The script provides examples of flowers with petal counts that are Fibonacci numbers, such as lilies with three petals and daisies with 34, 55, or 89 petals, illustrating the natural occurrence of these mathematical patterns.

💡DNA

Deoxyribonucleic Acid (DNA) is the molecule that carries genetic information in living organisms. The script reveals that the Golden Ratio and Fibonacci Sequence can be found within the structure of DNA, such as the dimensions of the DNA molecule and the arrangement of its components, indicating a deep connection between mathematics and biology.

💡Music

The script discusses the presence of the Golden Ratio in 'Music,' suggesting that its use in the timing and pacing of compositions can make them more pleasing to the ear. It cites examples of classical music pieces by Beethoven and Mozart that are believed to incorporate the Golden Ratio, indicating its influence across different artistic domains.

💡Human Facial Construction

The 'Human Facial Construction' is described in the script as being influenced by the Golden Ratio, which is believed to contribute to facial symmetry and attractiveness. The Golden Spiral is said to shape the human face in a way that is considered aesthetically pleasing, and this principle is applied in fields such as plastic surgery and facial recognition software.

💡Advertising

In the context of 'Advertising,' the script explains that the Golden Ratio is used to create appealing visuals and designs that attract potential customers. It is found in logos, website layouts, and even the selection of models for advertisements, demonstrating the practical application of this mathematical principle in marketing strategies.

💡Anatomy of Living Things

The 'Anatomy of Living Things' is shown in the script to exhibit proportions that adhere to the Golden Ratio, particularly in the human body as depicted in Leonardo da Vinci's 'Vitruvian Man.' The video highlights how various body measurements and the number of appendages relate to the Golden Ratio, suggesting a fundamental pattern in biological structures.

Highlights

The Golden Ratio, approximately 1.618, is a proportion found in various aspects of life based on the Fibonacci Sequence.

The Fibonacci Sequence is a series where each number is the sum of the two preceding ones, such as 1, 1, 2, 3, 5, 8.

The Golden Ratio is evident in architecture, art, music, and nature, including the Great Pyramids of Giza.

The geometry of the Great Pyramids and their orientation suggest intentional use of the Golden Ratio.

The Taj Mahal incorporates the Golden Ratio in its architecture, showcasing Indian contributions to mathematics.

Leonardo da Vinci utilized the Golden Ratio in his art, including the Mona Lisa and the Vitruvian Man.

Spiral galaxies, including the Milky Way, exhibit the Golden Ratio in their structure and expansion.

The Fibonacci Sequence is observed in the arrangement of flower petals and the spiral patterns of plants.

DNA structure contains the Golden Ratio and Fibonacci Numbers in its measurements and molecular geometry.

Musical compositions, such as Beethoven's Fifth Symphony, apply the Golden Ratio for aesthetic appeal.

Facial attractiveness is often associated with proportions that align with the Golden Ratio.

Advertising leverages the Golden Ratio to create appealing visuals and effective brand logos.

The human body's proportions, as depicted in the Vitruvian Man, reflect the Golden Ratio.

The Golden Ratio is present in the anatomy of various living organisms, including segmented bodies of insects.

The Golden Ratio's prevalence suggests a universal principle in design, from galaxies to DNA.

The video invites viewers to explore and share personal examples of the Golden Ratio's influence in daily life.

Transcripts

- You've probably heard of the golden rule,

and you might even be using it in your daily life,

but there is another little bit of gold

that is playing a role in your daily life

that you might not know about.

The Golden Ratio, or 1.618.

The Golden Ratio is a proportion

based on the Fibonacci Sequence and can be found

almost anywhere in life, probably even in the room

where you are sitting.

For those who do not know the Fibonacci Sequence,

it's a series of numbers where each number

is the sum of the two previous numbers.

For example, one, one, two, three, five, eight

is a basic example of the Fibonacci Sequence.

The sums are all called Fibonacci Numbers.

These special little numbers can be found in architecture,

art, music, and even in nature.

Here we will explore some of the many places

where you can find the Golden Ratio,

and the Fibonacci Sequence.

- Amazing!

Number 10: the Great Pyramids of Giza.

The ancient Egyptians have intrigued the human race

for thousands of years, possibly the most famous

point of interest in Egyptian culture

are The Great Pyramids of Giza.

The most popular question about the pyramids

is how they were built, but maybe the question

should be why they were built,

specifically why they were

built in such a specific orientation.

When ariel photographs first became available

of the Great Pyramids, it became apparent

that there is a pattern to they're placement.

The first place that the ratio is seen

is in the geometry of the pyramids.

When applying the Pythagorean Theorem

to the height of the base of one of he internal triangles,

you will notice that the hypotenuse is just .025% off

of a perfect Golden Ratio triangle,

otherwise known as a Kepler Triangle.

While this could be a happy accident,

we are guessing that these ancient mathematicians

were attempting to be as exact as possible

when dragging these eight ton blocks across the desert.

The next place you can find the golden ratio

in the pyramids is their placement next to each other.

Most of the pyramids have been placed

in the exact orientation of a phi matrix grid,

which is based on the Golden Ratio.

This can be seen in the Pyramids of Queens,

and the Pyramids of Khufu and Khafre.

While the Golden Ratio can be found in many more places

in the complex of the pyramids, maybe the most astounding

can only be seen from above.

When drawing lines through the base corners

of the pyramids, you will notice

that they begin to intersect at a specific point,

mathematicians then began to notice a familiar shape

form when looking at the illustration from above.

The spiral shape hat is seen in so many Golden Ratio

related figures is seen once again in the placement

and orientation of the Great Pyramids of Giza.

Just to clarify, a golden spiral is a spiral

that grows by a factor of Phi as it increases in size.

These spirals can also be fit into the Phi matrix.

Number nine: The Taj Mahal.

One more wonder of the world that successfully implemented

the Golden Ration into it's architecture, the Taj Mahal.

It's no big secret that the Indian people have made

a great contribution to the field of mathematics,

so it should come as no surprise that their

most famous building is rich with mathematical references.

The first and most obvious place you see the Golden Ratio

on the Taj Mahal is on it's rectangular base

where the Phi matrix can be seen.

In addition to this, you can see other buildings

on the Taj Mahal premises that also implement

the Phi matrix into it's architecture.

Number eight: The Mona Lisa.

One of the most famous pieces of art in human history

is also one of the most famous examples

of the use of the Golden Ratio, the Mona Lisa.

Leonardo Da Vinci was a polymath who held talents

is subjects from architecture to astronomy.

What is best known for his contributions to art

in many of his works he was able to combine his talents

in mathematics with his artistic ability.

The Mona Lisa was no exception.

Throughout the picture, you can draw the Golden Rectangle,

for example the space between her eyes can be divided

into a Golden Rectangle, meaning that a Golden Rectangle

exists around her head, furthermore her entire body

can be fit into a Golden Rectangle.

You can even draw a Golden Spiral

from the center of her nose over her head

and through her shoulders.

Many of Da Vinci's works where the Golden Ratio

can be found is within the famous Vitruvian Man,

as well as The Last Supper.

Nowadays, artists look to incorporate the Golden Ratio

within their works in an attempt to emulate

the masterpieces of Renaissance.

Number seven: Spiral Galaxies.

The examples of the Fibonacci Sequence

and the Golden Ratio that we have provided so far

may lead you to think that this is some man made

and contrived conspiracy theory put forth by

mathematicians around the world,

but when we look to nature, we can see

that this is no accident.

It almost seems divine.

One of the most astounding places that we find

the Golden Ration in nature is at it's most grand scale.

The cosmos.

The Golden Spiral can be seen across the universe

in the shape of spiral galaxies,

and even in our milky way galaxy.

If you look at our galaxy, you can calculate

the familiar logarithmic growth by a factor of phi

as the galaxy expands, but the galactic implementation

of the Golden Ratio doesn't end here.

The shape of the earth and moon are also

related to the Golden Ratio.

When placed next to each other,

the radius of the moon plus the radius of the earth

equal the square root of phi.

In addition to this, the triangle that can be drawn

using the radius of the earth as the base

will create a triangle with the hypotonus measuring phi.

Another example of this Golden Spiral that can only be seen

from space is the shape of hurricanes.

Loot at any satellite picture of a hurricane,

and you can once again see the spiral form

in the raging winds of nature's most destructive force.

Number six: Flower Petals.

One of the most famous examples where the Fibonacci Series

and the Golden Ratio are seen in nature is within plants.

The Fibonacci Series can often be found in

the number of seeds or petals in many flowers

and the Golden Spiral can be found

within the shape of the entire flower.

Many flower petals come in quantities that are

Fibonacci Numbers, for example, lilies and irises

have three petals, butter cups have five peals,

and daisies can be found with 34, 55, and 89 petals.

If you look at pine cones and pineapples

from the bottom, you will notice that spines

of these two plants are oriented in clockwise,

and counterclockwise spirals.

The number of lines that make these spirals

are both Fibonacci numbers.

Flowers like roses have petal arrangements

that are golden ratios.

Nest time you're out in the garden, see how many

of your own plants follow these Fibonacci trends.

Number five: DNA.

The Golden Ratio is so abundant, it even exists

within your DNA.

You can find the Golden Ratio and the Fibonacci Sequence

in numerous places throughout the

Deoxyribonucleic Acid that makes up the human body.

Maybe the most obvious example is the actual

size of human DNA which measures 34 angstrom units long

by 21 angstrom units wide.

Both of these are Fibonacci Numbers.

It is even reported that a cross sectional view

of a DNA molecule shows a decagonal shape.

A decagon can be created by rotating a pentagon

by 36 degrees using the diagonals of the pentagons

that can be found in the fractals of

a DNA cross section, you can see the Golden Ratio

within the proportions of the diagonals and the sides

which would be phi, too.

Number four: music.

As we've said before, the Golden Ratio exists

across all subjects including music.

In some of the world's most popular pieces of music,

the golden ratio is seen in timing and pacing.

It is speculated that such pieces that implement

a Golden Ratio based pattern are more pleasing to the ear.

In most of these examples, mathematicians have broken songs

down by the Golden Mean, .618.

One of the most famous examples of this is

Beethoven's Fifth Symphony where the Golden Ratio

is used in multiple sections throughout the song.

Mozart has also been reported using the Golden Ratio

in many of his Sonatas.

Nowadays, you can find many composers,

musicians, artists, trying to implement the Golden Ratio

into their songs.

Number three: Human Facial Construction.

You've probably heard people saying

that humans are naturally attracted

to facial symmetry.

That symmetry is created by the Golden Ratio.

People often say the Mona Lisa is one of the most

pleasing paintings to look at,

and it's not coincidence that the Golden Spiral

that shapes her face is similar to that of most humans.

Many of the people that society deems most attractive

are the ones whose face most closely fits

into the phi matrix that a person's face fits into.

Plastic surgeons even use programs that help

to digitally alter someone's face so it better

fits the golden ratio.

This allows them to show their patients

how their face can be surgically altered

to look more attractive.

Facial recognition software is also based on

the Golden Ratio.

Number two: Advertising.

Companies are always looking for the best ways to advertise

to their potential customers, and the use

of the Golden Ratio is one of the most effective techniques.

We already know that faces that loosely

associated to the Golden Ratio, or found most appealing,

and you will find that these are the types of people

that are being sold.

You can also find the Golden Ratio in the logos

of some of the world's most popular brands,

such as Toyota or Pepsi.

The Golden Ratio is a powerful tool

for website and graphic designers.

Many of your favorite sites will be based

around the Golden Matrix, or the Golden Rectangle.

Take a look at the most basic templates

on some free web builders, and you will notice

that most of them share a very similar look

to the Golden Ratio grids.

Number one: Anatomy of Living Things.

Earlier we mentioned one of Leonardo Da Vinci's

most famous works, the Vitruvian Man.

One of the reasons that it was such a famous

drawing is because it outlined

the immense proportionality of the human body.

For example, you can see the Golden Ratio in

the length of the human head, the breadth of the shoulders,

the distance from the top of the head

to the fingertips, and even the length of your forearm.

The consistency of the number five in the human body

is also related to the Golden Ratio.

We have five appendages on our torso,

two arms, two legs, one head.

On each of our arms and legs, we have

five appendages in fingers and toes.

On our head we have five protruding

appendages in two ears, two eyes, and a nose.

Finally, as we all know, we have five senses.

Other groups in the Animal Kingdom also have

body proportions based on the Golden Rule.

The most visibly obvious example

are insects and their segmented bodies.

From the galaxies above us, right down to our DNA,

you can find the Golden Ratio in almost every aspect.

There are plenty of other examples out there

of how the Golden Ratio plays a role

in your day to day activities, but we just listed

some of our favorites.

Let us know some of your favorite ways

the Golden Ratio affects your life in the comments below.

Thanks for watching.