What is the Fibonacci Sequence & the Golden Ratio? Simple Explanation and Examples in Everyday Life

Summary

The Fibonacci sequence, a series where each number is the sum of the two preceding ones, is found in nature, art, and architecture. This sequence, named after Leonardo Fibonacci, has been traced back to ancient Indian texts and was popularized in the West through his book 'Liber Abaci.' It's closely related to the golden ratio (approximately 1.618033), which is seen in various natural forms like seashells and in human-made structures. The golden spiral, a logarithmic spiral expanding by the golden ratio with each turn, is another manifestation of these mathematical concepts, observed in phenomena like ocean waves and in the works of artists like Salvador Dali.

Takeaways

• 🌼 Many flowers have a number of petals that are Fibonacci numbers, such as 3, 5, 8, 13, 21.
• 🌱 The arrangement of leaves on cacti and seeds in sunflowers often follows the Fibonacci sequence in both left- and right-handed spirals.
• 🤲 The human hand, with 2 hands of 5 fingers each, and each finger divided into 3 parts, exhibits Fibonacci numbers in its structure.
• 📐 The Fibonacci sequence is a series where each number is the sum of the two preceding ones, starting with 0, 1, 1, 2, 3, and so on.
• 🧮 The sequence is named after Leonardo Fibonacci, a medieval mathematician who introduced the Hindu-Arabic numeral system to the Western world through his book Liber Abaci.
• 📘 The Fibonacci sequence has been known since at least the 6th century in ancient Indian texts, predating Fibonacci's work.
• 🔢 The sequence is mathematically represented by the formula F(n) = F(n-1) + F(n-2) for n > 1.
• 🌟 The Fibonacci sequence is closely related to the golden ratio (phi, ϕ), approximately 1.618033, which is a ratio of two consecutive Fibonacci numbers as they get larger.
• 🐚 The golden spiral, formed by applying the golden ratio as a growth factor, is seen in many natural phenomena like seashells, ocean waves, and hurricanes.
• 🏛 The golden ratio and spirals are used in art, architecture, and design, including works by Salvador Dali and the architectural principles of Le Corbusier.

Q & A

What is the significance of the Fibonacci sequence in nature?

The Fibonacci sequence is significant in nature as it appears in the arrangement of petals, leaves, and seeds in various plants, often following the sequence's pattern. This can be observed in the number of petals on a flower, the arrangement of leaves on cacti, and the spiral patterns in sunflowers.

How is the Fibonacci sequence defined mathematically?

Mathematically, the Fibonacci sequence is defined by the formula F(n) = F(n-1) + F(n-2), where n is greater than 1. This means each number in the sequence is the sum of the two preceding numbers, starting with 0 and 1.

Who is credited with popularizing the Fibonacci sequence in the Western world?

Leonardo Pisano, also known as Fibonacci, is credited with popularizing the Fibonacci sequence in the Western world through his book Liber Abaci, published in 1202.

What is the connection between the Fibonacci sequence and the golden ratio?

The golden ratio is an offshoot of the Fibonacci sequence. As you progress through the sequence, the ratio of consecutive Fibonacci numbers converges to the golden ratio, which is approximately 1.618033.

What is the golden ratio, and how is it represented?

The golden ratio is a special number, often represented by the Greek letter phi (ϕ), that arises when a line is divided in such a way that the ratio of the whole line to the longer part is equal to the ratio of the longer part to the shorter part. It is approximately equal to 1.618033.

How does the golden ratio manifest in geometry?

In geometry, the golden ratio manifests as a growth factor in logarithmic spirals known as golden spirals. These spirals expand by a factor of phi with each quarter turn.

Where can the golden spiral be found in nature?

Golden spirals can be found in various natural forms such as seashells, ocean waves, hurricanes, flower buds, snail shells, and spider webs.

How have artists and architects utilized the golden ratio or spiral?

Artists like Salvador Dali have explicitly used the golden ratio in their works, such as The Sacrament of the Last Supper. Architects, including Le Corbusier, have incorporated the golden ratio into their designs for aesthetic and functional purposes.

What is the significance of the golden ratio in architecture?

In architecture, the golden ratio is used to create aesthetically pleasing and harmonious designs. It is believed to contribute to the visual balance and proportion of buildings and structures.

Can you provide an example of how Fibonacci numbers relate to the human body?

Fibonacci numbers relate to the human body in various ways, such as the number of fingers on each hand (5), the number of bones in each finger (3), and the proportions of bone lengths in the hand, which often follow Fibonacci numbers.

Why is the Fibonacci sequence considered ubiquitous in our everyday lives?

The Fibonacci sequence is considered ubiquitous because it appears in various aspects of mathematics, nature, art, and architecture, often associated with patterns of growth and proportion that are aesthetically pleasing and functionally efficient.

Outlines

00:00

🌼 The Ubiquity of Fibonacci Numbers in Nature

This paragraph discusses the prevalence of Fibonacci numbers in the natural world, particularly in the number of petals on flowers, the arrangement of seeds in sunflowers, and the spiral patterns in cacti. It also touches on the human hand's structure, which includes Fibonacci numbers in the number of fingers and bone lengths. The Fibonacci sequence is introduced as a series where each number is the sum of the two preceding ones, starting with 0 and 1. The historical context is provided by mentioning Leonardo Pisano, also known as Fibonacci, who introduced the Hindu-Arabic numeral system to the Western world through his book 'Liber Abaci' in 1202, which also contains the earliest known description of the Fibonacci sequence outside of India.

Keywords

💡Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. This sequence is central to the video's theme as it appears in nature, art, and architecture, and is known for its mathematical significance. Examples from the script include the number of petals on a flower, spirals in cacti, and the bone structure in the human hand.

💡Fibonacci Numbers

Fibonacci Numbers are the individual numbers in the Fibonacci Sequence. These numbers frequently appear in natural patterns and are illustrated in the video through examples like the number of seeds in a sunflower and the arrangement of leaves on a plant. They are a core concept, highlighting the connection between mathematics and nature.

💡Golden Ratio

The Golden Ratio is a mathematical ratio, approximately 1.618, that is closely related to the Fibonacci Sequence. In the video, it is described as a ratio that appears when the ratio of two quantities is equal to the ratio of their sum to the larger quantity. The Golden Ratio is linked to the Fibonacci Sequence and is seen in various natural forms, artistic works, and architectural designs, such as seashells and Le Corbusier's Modulor system.

💡Leonardo of Pisa (Fibonacci)

Leonardo of Pisa, also known as Fibonacci, was a medieval mathematician who introduced the Fibonacci Sequence to the Western world through his book 'Liber Abaci.' The video credits him with popularizing the sequence and the Hindu-Arabic numeral system in Europe. His contributions are fundamental to the video's exploration of the sequence's widespread presence in nature and human design.

💡Liber Abaci

'Liber Abaci' is a book written by Fibonacci in 1202, which introduced the Hindu-Arabic numeral system to Europe. The book is significant in the video's narrative as it contains the earliest known description of the Fibonacci Sequence outside India. It highlights Fibonacci's role in advancing mathematical knowledge in the Western world.

💡Hindu-Arabic Numeral System

The Hindu-Arabic Numeral System is a positional number system that Fibonacci introduced to Europe, as described in his book 'Liber Abaci.' This system, more efficient than Roman numerals, is emphasized in the video as a significant mathematical advancement that Fibonacci helped popularize, contributing to the spread of the Fibonacci Sequence.

💡Golden Spiral

The Golden Spiral is a logarithmic spiral that widens by a factor of the Golden Ratio for every quarter turn. The video discusses this spiral as a geometric manifestation of the Fibonacci Sequence and Golden Ratio, with examples like seashells and hurricanes illustrating its occurrence in nature.

💡Logarithmic Spiral

A Logarithmic Spiral is a curve that appears frequently in nature and is mathematically related to the Golden Ratio. The video mentions it in the context of the Golden Spiral, explaining how the Fibonacci Sequence and the Golden Ratio contribute to the formation of such spirals in natural phenomena like ocean waves and galaxies.

💡Divine Proportion

The Divine Proportion is another term for the Golden Ratio, emphasizing its perceived beauty and perfection in nature and art. The video uses this term to describe the ubiquitous presence of the Golden Ratio in natural forms, human anatomy, and artistic creations, suggesting a mystical quality to its recurrence.

💡Le Corbusier

Le Corbusier was a Swiss-French architect who applied the Golden Ratio in his architectural designs, particularly in his Modulor system. The video highlights his work as an example of the Golden Ratio's influence on modern architecture, showing how this mathematical concept has practical applications in design.

Highlights

The number of petals on many flowers is a Fibonacci number, such as 3, 5, 8, 13, or 21.

Cacti leaves and sunflower seeds are arranged in left- and right-handed spirals that often correspond to Fibonacci numbers.

Humans have 2 hands, each with 5 fingers, and each finger is divided into 3 parts, all of which are Fibonacci numbers.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1.

Mathematically, the Fibonacci sequence is represented by the formula: F(n) = F(n-1) + F(n-2), where n > 1.

Leonardo Pisano, known as Fibonacci, introduced the Hindu-Arabic numeral system to the Western world through his book 'Liber Abaci'.

The Fibonacci sequence has roots in ancient Indian mathematics, documented by mathematicians as early as the 6th century.

The Fibonacci sequence appears not only in mathematics but also in various forms in nature and daily life.

The golden ratio, represented by the Greek letter phi (ϕ), is closely related to the Fibonacci sequence.

If the ratio of two quantities equals the ratio of their sum to the larger quantity, they have a golden ratio, approximately 1.618033.

A golden spiral, which widens by a factor of phi for every quarter turn, is a geometric manifestation of the golden ratio.

Examples of the golden spiral in nature include seashells, ocean waves, hurricanes, flower buds, snail shells, and spider webs.

The artist Salvador Dali used the golden ratio in his painting 'The Sacrament of the Last Supper'.

Architect Le Corbusier applied the golden ratio in his Modulor system for architectural proportion.

The golden ratio, also known as the divine proportion, is found in various natural and man-made structures, from snail shells to galaxies.

Transcripts

Have you ever counted the  number of petals on a flower?  

If you started counting them on a flower whose  petals are all intact, you’d most likely notice  

that the number of petals is a Fibonacci number,  that is, a number from the Fibonacci sequence.  

Many flowers have 3,5,8, 13, 21 or more petals. The leaves of cacti and seeds of sunflowers are  

arranged in both left- and right-handed spirals.  The numbers of seeds or leaves in these spirals  

are generally in the Fibonacci sequence. We have 2 hands, both of which have 5 fingers,  

and each finger is divided into 3 parts.  All of these numbers are Fibonacci numbers.  

Moreover, the lengths of bones in  hand are all Fibonacci numbers too.  

So what’s the Fibonacci sequence anyway?  And why is it seemingly everywhere?  

The Fibonacci sequence is a series  of numbers in which a given number  

is the addition of the two numbers before it. So, if you start with 0, the next number  

will be 1, followed by 1, followed  by 2, followed by 3 and so on.  

(not to be spoken in audio) 0, 1,  1, 2, 3, 5, 8, 13, 21, 34, 55…..  

As you can see, every number in this series or  sequence is obtained by adding the two preceding  

numbers. This simple series of numbers is  referred to as the Fibonacci sequence for  

Fibonacci series. And individual numbers in this  sequence are often called Fibonacci numbers.  

Mathematically, the fibonacci sequence  is represented with this formula  

F(n) = F(n-1) + F(n-2) (not  to be spoken in audio)  

where n>1 You can use this expression  

to find any ‘n’th digit in the sequence. This fascinating sequence is widely associated  

with the mathematician, Leonardo Pisano, aka  Fibonacci. He hailed from the Republic of Pisa,  

which is why he is also known as Leonardo  of Pisa and was known as one of the most  

talented mathematicians of the Middle ages. At the time, Europeans were still using Roman  

numbers while Hindu-Arabic mathematicians  were using a different number system  

which was more robust and efficient. Fascinated by  the brilliance of the Hindu-Arabic numeral system,  

Fibonacci brought them to the western world in  1202 through his now-famous book Liber Abaci.  

In the book, he reviewed and compared the  Hindu-Arabic numeral system with other systems,  

such as Roman numerals, and described  how using the Hindu-Arabic system made  

calculations faster and easier. Although his book contains the  

earliest known description of the  Fibonacci sequence outside of India,  

it has been described in ancient Indian texts  by mathematicians as early as the 6th century.  

So, what makes the Fibonacci sequence so special? To the uninitiated, it may just seem like a series  

of numbers, but the Fibonacci sequence has been  discovered and rediscovered in various forms,  

not only in mathematics, but also  in nature and our everyday lives.  

There’s another exciting offshoot of the  Fibonacci sequence - the golden ratio.  

Suppose you have two quantities, A  and B, wherein A is greater than B.  

Now, add A and B and divide the sum by  A. If this ratio comes out to be equal  

to the ratio of A and B, then you’d  say that A and B have a golden ratio.  

It’s represented by the Greek letter phi ( ϕ ). Write down the Fibonacci sequence on a piece  

of paper and calculate the ratio using  this formula. You will notice that all  

Fibonacci numbers have the golden ratio,  the value of which is close to 1.618033…  

In geometry, when the golden ratio is applied  as a growth factor, you get a special kind of  

logarithmic spiral known as a golden spiral.  Simply put, a golden spiral gets wider by a  

factor of phi for every quarter turn it makes. You’ll find examples and manifestations  

of golden ratio and golden spirals in  countless places in your everyday life.  

Seashells are one of the most common examples  of the golden spiral in nature. Ocean waves,  

hurricanes, flower buds, snail shells, spider webs  are some of the many naturally-occurring examples. 

Many artists use the golden ratio or spiral  in their creative works. Legendary painter  

Salvador Dali explicitly used the ratio in his  masterpiece The Sacrament of the Last Supper.  

Architects often use the golden ratio in designing  buildings and massive other massive structures.  

Popular Swiss-French architect Le Corbusier,  who is widely acclaimed as one of the pioneers  

of modern architecture, explicitly used  the golden ratio in his Modulor system  

for the scale of architectural proportion. From snail shells to flowers, from bananas to  

the inside of the human ear, from large,  iconic buildings to galaxies, the golden  

ratio can be seen everywhere. This is probably  why it’s also known as the divine proportion.  

There are so many places where you can find the  golden ratio or the Fibonacci numbers that it’s  

impossible to list all of them in one place and  be done with it, because the list is ever-growing.  

What other examples of the Fibonacci sequence or  the golden ratio have you observed in your life?  

Tell us in the Comments below!