Fibonacci Sequence and Golden Ratio || Mathematics in the Modern World

Summary

In this video, Ram explores the fascinating presence of mathematics in nature, specifically the Fibonacci sequence, which is often found in the arrangement of petals and the growth patterns of various natural phenomena. The script delves into the historical roots of the sequence, its mathematical properties, and its prevalence in art, architecture, and even human biology. The golden ratio, derived from the Fibonacci sequence, is highlighted for its aesthetic appeal and its use in creating harmonious proportions in diverse fields, from Renaissance art to modern-day photography apps. The video also touches on the practical applications of these mathematical concepts in finance and the natural world, showcasing their enduring significance.

Takeaways

• 🧠 Mathematics is deeply intertwined with nature, with patterns like the Fibonacci sequence found in various natural phenomena.
• 🌼 The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 or 1.
• 🌐 The sequence was studied by ancient Indian mathematicians and later popularized in Europe by Leonardo Fibonacci.
• 📚 Fibonacci's influence is significant, as he contributed to the spread of the Hindu-Arabic numeral system in the Western world through his book, 'Liber Abaci'.
• 🐰 The Fibonacci sequence can be illustrated with a hypothetical problem involving the population growth of rabbits, leading to a prediction of 233 pairs or 466 rabbits after a year.
• 📏 The Fibonacci sequence can be visualized geometrically, forming a spiral that is prevalent in nature, art, architecture, and even the human body.
• 📐 Dividing two consecutive Fibonacci numbers yields the Golden Ratio, approximately 1.618, which is considered aesthetically pleasing and is found in various natural and man-made structures.
• 🎨 The Golden Ratio has been used in art and architecture to create balanced and harmonious compositions, as seen in the works of Renaissance masters like Leonardo da Vinci and Michelangelo.
• 🎶 Even music has been influenced by the Golden Ratio, with evidence of its use in the compositions of famous classical musicians like Debussy, Mozart, Beethoven, and Bach.
• 🌳 Nature exhibits the Fibonacci sequence and Golden Ratio in the branching patterns of trees, spirals on snails, and the structure of galaxies.
• 📈 The Fibonacci sequence and Golden Ratio have practical applications in finance, assisting analysts in predicting and understanding market trends.

Q & A

What is the main idea presented in the video script?

The main idea is that mathematics, specifically the Fibonacci sequence and the Golden Ratio, are inherently present in nature and have been utilized in various fields such as art, architecture, and even biology.

What is the Fibonacci sequence and how is it formed?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. It appears in various natural phenomena and has been studied by mathematicians for centuries.

Who are some of the early investigators of the Fibonacci sequence mentioned in the script?

Early investigators of the Fibonacci sequence include Indian mathematicians Pingala, Virahanka, and Hemachandra.

How was the Fibonacci sequence popularized in Europe?

The Fibonacci sequence was popularized in Europe by Leonardo of Pisa, also known as Leonardo Fibonacci, who introduced it through his book 'Liber Abaci' or 'Book of Calculation'.

What problem did Fibonacci solve in his book that led to the discovery of the Fibonacci sequence?

Fibonacci solved a problem involving the idealized growth of a rabbit population, which led to the discovery of the Fibonacci sequence as the solution.

What is the significance of the number 233 in the context of the rabbit population problem?

The number 233 represents the total number of rabbit pairs (or 466 individual rabbits) after one year, based on the assumptions of the rabbit population problem posed by Fibonacci.

What is the Golden Ratio and how is it related to the Fibonacci sequence?

The Golden Ratio, approximately equal to 1.618, is a mathematical concept that describes a proportional relationship between two quantities. It is related to the Fibonacci sequence because the ratio of consecutive Fibonacci numbers approaches the Golden Ratio as the numbers get larger.

How is the Golden Ratio used in art and architecture?

The Golden Ratio is used to create aesthetically pleasing and harmonious proportions in art and architecture. It has been employed by artists like Leonardo da Vinci, Michelangelo, and in architectural structures like the Parthenon and the Taj Mahal.

What is the significance of the Golden Ratio in the human body?

The Golden Ratio is believed to be present in the proportions of the human body, with certain bone structures and measurements conforming to this ratio, which is thought to contribute to perceived beauty and harmony.

How is the Fibonacci sequence and the Golden Ratio applied in the field of finance?

In finance, the Fibonacci sequence and the Golden Ratio are used by analysts to predict and understand market trends and patterns, providing a mathematical approach to investment strategies.

What is the connection between the Fibonacci sequence and the background music in the video?

The background music in the video is composed based on the Fibonacci sequence, illustrating the application of mathematical patterns in creative fields like music.

Outlines

00:00

📚 Introduction to the Fibonacci Sequence in Nature

The first paragraph introduces the concept that mathematics, specifically the Fibonacci sequence, is not just an abstract concept but is deeply rooted in nature. The sequence is characterized by each number being the sum of the two preceding ones, starting from 1 and 2. The speaker, Ram, suggests that this sequence is prevalent in the number of petals in flowers and other natural phenomena. The paragraph also mentions that the sequence was studied by ancient Indian mathematicians and was popularized in Europe by Leonardo Fibonacci, who is known for his influential work on the Hindu-Arabic numeral system. The summary of a problem involving the growth of a rabbit population using the Fibonacci sequence is provided, leading to the conclusion that after a year, there would be 466 rabbits.

05:02

🎨 The Golden Ratio and Its Ubiquity in Art, Architecture, and Nature

The second paragraph delves into the concept of the golden ratio, approximately 1.618, and its significance in creating aesthetically pleasing proportions. The golden ratio is found by dividing two consecutive Fibonacci numbers and is represented by the Greek letter phi (Φ). The speaker discusses how this ratio has been used historically in art by renowned artists like Leonardo da Vinci and in architecture in structures such as the Parthenon and the Taj Mahal. It also touches on the golden ratio's presence in music, the Bible, and even in the human body and DNA. The paragraph concludes by mentioning the practical applications of the Fibonacci sequence and the golden ratio in various fields, including finance and photography, and notes that the background music in the video is also derived from the Fibonacci sequence.

Keywords

💡Matu klassan

Matu klassan is presumably the name of the educational channel or series that the video script belongs to. It is the context in which the host, Ram, introduces the topic of mathematics and its natural occurrence, setting the stage for the exploration of mathematical patterns in nature.

💡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. In the video, it is highlighted as a pattern found in nature, such as the number of petals on a flower, and is deeply connected to the theme of mathematics being inherent in the natural world.

💡Golden ratio

The golden ratio, approximately 1.618, is a mathematical concept that represents an aesthetically pleasing proportion found in art, architecture, and nature. The video explains that this ratio is derived from consecutive Fibonacci numbers and is used to illustrate the harmonious balance present in various creations, including human-made structures and natural phenomena.

💡Leonardo Fibonacci

Leonardo Fibonacci, also known as Leonardo of Pisa, is credited with popularizing the Hindu-Arabic numeral system in the Western world through his book 'Liber Abaci'. In the script, he is noted for introducing the Fibonacci sequence, which he used to solve a problem involving the idealized growth of a rabbit population.

💡Golden spiral

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every quarter-turn it makes. The video describes how this spiral can be visualized using square tiles and is found in various natural forms, such as the arrangement of leaves or the spirals of galaxies.

💡Hindu-Arabic numeral system

The Hindu-Arabic numeral system is the base-10 number system that includes the digits from 0 to 9 and is the most widely used numeral system in the world today. The video mentions that Leonardo Fibonacci played a significant role in popularizing this system in Europe.

💡Pingala, Virahanka, and Hema Chandra

These Indian mathematicians are mentioned in the script as early investigators of the Fibonacci sequence-like patterns. Their work predates the European popularization of the sequence, indicating the global history and influence of mathematical thought.

💡Liber Abaci

Liber Abaci, or 'The Book of Calculation', is the book written by Leonardo Fibonacci that introduced the Hindu-Arabic numeral system to Europe. The video script uses it as an example of how the Fibonacci sequence was used to solve a problem related to rabbit population growth.

💡Vitruvian Man

The Vitruvian Man is a drawing by Leonardo da Vinci that represents the ideal human proportions based on the golden ratio. The video script refers to this work to illustrate the application of the golden ratio in human anatomy and the pursuit of aesthetic perfection.

💡Divine Proportion

Divine Proportion is a mathematical textbook written by Luca Pacioli, which Leonardo da Vinci illustrated. The video mentions this book in the context of the golden ratio's significance in art and human body proportions, emphasizing the mathematical underpinnings of beauty and harmony.

💡Fibonacci in nature

The script discusses how the Fibonacci sequence appears in various natural phenomena, such as the arrangement of leaves, the spirals of snail shells, and the branching patterns of trees. This concept reinforces the video's theme of mathematics being a fundamental aspect of the natural world.

Highlights

Mathematics is present in nature, and mathematicians translate it using numbers.

Nature often favors specific numbers like 1, 2, 3, 5, 8, and 13, forming the Fibonacci sequence.

The Fibonacci sequence follows a pattern where each number is the sum of the two preceding ones.

Indian mathematicians like Pingala, Virahanka, and Hema Chandra had early investigations into the pattern.

Leonardo Fibonacci popularized the sequence in Europe through his book 'Liber Abaci'.

Fibonacci's work led to the popularization of the Hindu-Arabic numeral system in the Western world.

The Fibonacci sequence can be visualized using square tiles to form a spiral, seen in nature and human-made structures.

Dividing two consecutive Fibonacci numbers results in the Golden Ratio, approximately 1.618.

The Golden Ratio is used to describe the complementary relationship between two figures in perfect proportion.

Many famous artists and architects, including Leonardo da Vinci, used the Golden Ratio in their works.

The Golden Ratio is found in various natural phenomena, including the branching of trees and the spirals on snails.

The Fibonacci sequence and Golden Ratio have practical applications in fields like finance and photography.

The background music in the video is also derived from the Fibonacci sequence.

The Fibonacci sequence can be used to model the growth of populations, such as rabbits, over time.

The sequence can be extended to model the number of rabbits after a year, resulting in 466 rabbits.

The Golden Ratio is approximately equal to the Greek letter phi (Φ) and is used to calculate perfect spirals and figures.

Psychologists have noted that attractive personalities often conform to the Golden Ratio in their facial and body structures.

The Fibonacci sequence and Golden Ratio are integral to understanding market patterns in the finance sector.

Transcripts

[Music]

hello everyone

my name is ram and welcome to another

video of matu klassan

some people may think that mathematics

was made to torture our brains on

nonsense numbers but what if math

is already present in nature and

mathematicians are just translating it

to us

using symbols like numbers

have you ever counted the number of

petals in a flower

you might think that any number is

possible but you might be surprised

because

nature seems to favor a particular set

of numbers

like 1 2 3 5

8 and 13. it may seem a coincidence to

you

but these some sort of numbers form a

pattern

in the sequence the next number is found

by adding up the two numbers before it

for example 2 is found by adding the

numbers before it

one plus one equals two following the

same pattern

one plus two equals three two plus three

equals five three plus five

equals eight and five plus eight

equals thirteen this pattern

is called fibonacci sequence

early writing show that this pattern of

numbers was already investigated by some

indian mathematicians like

pingala virahanka and hema chandra

though this pattern was popularized in

europe by leonardo of pisa

also known as leonardo fibonacci thus

the name of the pattern

he's one of the most influential

mathematicians of the middle ages

because hindu arabic numeral system

which we also use today

was popularized in the western world

because of his book

liber abachi or book of calculation

in this book fibonacci posed and solved

a problem

involving the growth of a population of

rabbits based on

idealized assumptions the solution for

this problem

well as you guessed the fibonacci

sequence

now let's investigate the problem

further

if two newborn rabbits are put in a pen

how many rabbits will be in the pan

after a year

assume that rabbits always produced one

male

and one female offspring can reproduce

once every month can reproduce once they

are one month old and

they never die we start with the first

pair of newborn rabbits at the beginning

of the month

since they are too young to produce an

offspring

when they get old at the beginning of

the second month

we still have one pair this time

the rabbits are old enough to reproduce

so

at the start of the third month we will

have two

pairs on the fourth

month the old pair will produce another

pair

while the first set of offspring will be

old enough to

reproduce so on the fifth month

we will have five pairs of rabbits

by using the fibonacci sequence we will

have

144 pairs in the beginning of the 12th

month

however we need to identify the number

of rabbits

after a year so we need to continue the

sequence

up to the 13th term of the sequence

so the final answer to the problem is

233 pairs

or 466 rabbits

fibonacci sequence is a wonderful series

of numbers that could either start with

0

or 1. let's visualize these numbers

using square tiles i will start with

a one by one square then

another together they form a one by two

a rectangle

above that a two by two square

next to that is a three by three square

beneath that a five by five square

now if we continue to do this and

connect opposing diagonals continuously

it will reveal the fibonacci spiral

and this spiral could be seen a lot in

nature

architecture arts human body

and beyond going back to the rectangle

what if we're going to divide the two

dimensions

8 and 13 on this case notice that

it's just like dividing two consecutive

fibonacci numbers right

doing this up to the highest possible

pair of fibonacci numbers

will give us the golden ratio

the golden ratio is approximately equal

to

1.618 represented by the greek letter

p or phi in math the golden ratio is a

term

used to describe the relationship of two

figures

where the numbers seem to be in some

form of complementary ratio

if you have a number a and a lower

number b

then the two are in the golden ratio if

the quotient of these two numbers

somehow a near 1.618

so basically any ratio which comes close

to this value

is said to be perfectly proportioned

the golden ratio is commonly used to

calculate perfect spirals

triangles arcs and other figures for

example

a rectangle with a length of 8.09 inches

and a width of 5

inches is in the golden ratio because

8.09

divided by 5 is equal to 1.618

since the discovery of the golden ratio

many known individuals were inspired to

incorporate this magnificent number

to their greatest work and creation

during the renaissance

leonardo da vinci used the proportions

set forth by the golden ratio

to construct his masterpieces sandro

botticelli

michael michelangelo and others appear

to have employed this same technique

in their artwork in a music

evidence were found on the work of a

debussy

mozart beethoven bach and chauffen

to mans a marvelous architecture like

parthenon

taj mahal roman arches egyptian pyramids

eiffel tower notre dame cathedral and

many more were also built

based on this mathematical pattern even

in the bible in exodus chapter 25 verse

10 a god commands moses to build the ark

of the covenant

in which to hold his covenant with the

israelites

and even the ark which god commands nova

to build is based

on a variation of the same ratio

in the 1490s leonardo da vinci

illustrated a mathematical textbook

entitled

divine proportion by luca patioli

in this book along with his work the

vitruvian man

he pointed out the existence of the

golden ratio

in the human body this can be proven by

using the actual comparisons of the

measurements

of our bone structures and body

it even applies to our dna yep

shockingly true psychologists have also

noted that many celebrities and other

attractive personalities

have facial body and bone structures

that conform to the golden ratio

draenoraya in nature

aside from the branching patterns of

trees spirals on snails

animal horns and seed heads did you know

that honeybee colonies maintain a ratio

of males to females at 1 to 1.618

this number even extends to natural

occurrences

earth moon relationship solar system and

galaxies

you may even have it on your phone yes

because aside from the rule of thirds in

photography

fibonacci spiral is also being used

you might want to check that out it's

also available in app store and

google play so how is this even possible

probably some of you will say that they

have no practical use

but the fibonacci sequence and the

golden ratio

play a vital role in the finance sector

worldwide

like helping financial analysts predict

and understand

the market maybe those are just

coincidences

and don't mean anything to few

however coincidence or not these

patterns became

part of the world we are living today

and they help us unravel the mysteries

of

nature and before i forget

the background music i'm playing

it also came from the fibonacci sequence

[Music]