Fibonacci Sequence and Golden Ratio || Mathematics in the Modern World
Summary
TLDRIn 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
📚 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.
🎨 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.
Mindmap
Keywords
💡Matu klassan
💡Fibonacci sequence
💡Golden ratio
💡Leonardo Fibonacci
💡Golden spiral
💡Hindu-Arabic numeral system
💡Pingala, Virahanka, and Hema Chandra
💡Liber Abaci
💡Vitruvian Man
💡Divine Proportion
💡Fibonacci in nature
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]
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