Fibonacci Sequence
Summary
TLDRThis lecture delves into the Fibonacci sequence, a series of numbers where each term is the sum of the two preceding ones, starting from 0 and 1. Originating from the work of Leonardo Pisano, also known as Fibonacci, it has far-reaching implications in nature, art, and mathematics. The sequence appears in patterns like sunflower seeds and rabbit population growth, and is closely tied to the Golden Ratio, a mathematical constant found in various natural phenomena. The lecture illustrates how mathematics can quantify and predict natural occurrences, emphasizing its importance beyond problem-solving.
Takeaways
- 📚 Leonardo Pisano, known as Fibonacci, was an Italian mathematician who lived between 1170 and 1250 and is famous for the Fibonacci sequence.
- 🌟 The Fibonacci sequence starts with 0, 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
- 🌱 The sequence appears in nature, such as the spiral patterns in sunflowers, though not all sunflowers strictly follow this pattern.
- 🌼 Examples of Fibonacci in nature include the patterns in flowers like the mariposa lily, guava, melon, marigold, and even the arrangement of a banana's segments.
- 🔢 The squares of the Fibonacci numbers follow a pattern where the sum of the squares of the first n numbers equals the square of the (n+1)th Fibonacci number.
- 🐇 Fibonacci's rabbit problem illustrates the sequence by showing how the number of rabbit pairs increases each month based on the sum of pairs from the two previous months.
- 🎯 The Golden Ratio, approximately 1.618034, is closely related to the Fibonacci sequence, often appearing when a line is divided in a way that the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller part.
- 🌀 Many plants grow in spirals, often with the number of spirals being a Fibonacci number, resembling the Fibonacci spiral.
- 🔢 The formula for the Golden Ratio is given by (1 + √5) / 2, which is derived from the ratio of successive Fibonacci numbers.
- 🌐 Mathematics helps us understand patterns in nature and occurrences in our world, serving as a tool to quantify, organize, and predict phenomena.
Q & A
Who is Leonardo Pisano Bergoglio and what is his contribution to mathematics?
-Leonardo Pisano Bergoglio, also known as Fibonacci, was an Italian mathematician who lived between 1170 and 1250. He is best known for introducing the Hindu-Arabic numeral system to Europe and developing the famous Fibonacci sequence.
What is the Fibonacci sequence?
-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 goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
How does the Fibonacci sequence relate to nature?
-The Fibonacci sequence appears in nature in various patterns, such as the arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, or the pattern of a pinecone's bracts.
What is the significance of the Fibonacci sequence in sunflowers?
-Sunflowers often exhibit the Fibonacci sequence in the number of spirals, with common counts being 21 and 34. However, it's not a rule for all sunflowers, as some may not conform to the sequence.
Can you provide an example of how the Fibonacci sequence appears in other flowers besides sunflowers?
-The Fibonacci sequence can be found in the patterns of many flowers, such as the Mariposa lily, guava, melon, marigold, and even in the arrangement of a banana's segments.
What is the relationship between the Fibonacci sequence and the squares of its numbers?
-The sum of the squares of the first n natural numbers is equal to the square of the nth Fibonacci number. For example, 1^2 + 2^2 + 3^2 + 5^2 + 8^2 = 89, which is the square of 13.
What is the Fibonacci rabbit problem?
-The Fibonacci rabbit problem is a classic example created by Fibonacci to illustrate the sequence. It concerns the growth of a rabbit population where each month, each pair of mature rabbits produces a new pair, and no rabbits die. The number of rabbit pairs each month follows the Fibonacci sequence.
What is the Golden Ratio and how is it related to the Fibonacci sequence?
-The Golden Ratio is a mathematical constant found by dividing a line into two parts such that the ratio of the whole line to the longer part is the same as the ratio of the longer part to the shorter part. It is approximately 1.618. The Golden Ratio is closely approximated by successive Fibonacci numbers.
How does the Fibonacci sequence appear in the growth patterns of plants?
-Many plants grow in spirals, often with the number of spirals being a Fibonacci number. This can be seen in the arrangement of leaves, seeds, or fruits, resembling the Fibonacci spiral.
What is the significance of the Fibonacci sequence in mathematics and nature?
-The Fibonacci sequence is significant because it appears in various patterns in nature and can help explain occurrences and phenomena. It also demonstrates the interconnectedness of mathematics and the natural world.
Outlines
📐 Introduction to Fibonacci Sequence
The script introduces the Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. It discusses the historical context of the sequence's development by Leonardo Pisano, also known as Fibonacci, who lived between 1170 and 1250 in Italy. Fibonacci's work helped to spread the Hindu-Arabic numeral system across Europe, replacing Roman numerals. The script explains the sequence and its progression, and how it appears in nature, particularly in the patterns of sunflowers and other plants. It also touches on the mathematical concept of squaring the Fibonacci numbers and the relationship to the golden ratio.
🌱 Fibonacci in Nature and the Golden Ratio
This paragraph delves into the occurrence of the Fibonacci sequence in nature, exemplified by the patterns found in sunflowers, lilies, and other plants. It mentions that while the Fibonacci sequence is often associated with the number of spirals in sunflowers, not all sunflowers conform to this pattern. The script also explores the concept of the golden ratio, which is closely related to the Fibonacci sequence. The golden ratio is described as a special number that arises when a line is divided into two parts such that the ratio of the whole length to the longer part is equal to the ratio of the longer part to the shorter part. The golden ratio is approximately 1.618, and it is found in many natural phenomena and artistic compositions.
🔢 Mathematics and Its Role in Understanding the World
The final paragraph of the script emphasizes the importance of mathematics in understanding patterns and occurrences in nature and life. It suggests that mathematics is not just about solving equations but also about comprehending the 'why' behind phenomena. The paragraph concludes with a generalization about mathematics being a tool to quantify, organize, and predict occurrences in the world, making life easier. It wraps up the lecture on the Fibonacci sequence and invites the audience to look forward to the next lecture.
Mindmap
Keywords
💡Fibonacci Sequence
💡Leonardo Pisano (Fibonacci)
💡Hindu-Arabic Numerals
💡Golden Ratio
💡Spiral Patterns
💡Sunflowers
💡Mariposa Lily
💡Mathematical Patterns
💡Rabbit Problem
💡Quantify
💡Organize
Highlights
Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones.
Leonardo Pisano, also known as Fibonacci, introduced the sequence in Europe.
Fibonacci sequence appears in nature, such as in the arrangement of sunflower seeds.
Not all sunflowers follow the Fibonacci sequence perfectly.
The sequence can be found in various flowers like marigolds, lilies, and bananas.
The Fibonacci sequence can be represented as a series of squares.
The sum of the squares of the Fibonacci numbers equals the square of the next number in the sequence.
Fibonacci sequence is related to the growth pattern of rabbits.
The Golden Ratio is closely associated with the Fibonacci sequence.
The Golden Ratio is approximately 1.618 and is found in various natural patterns.
Many plants grow in spirals, often with the number of spirals being a Fibonacci number.
The Fibonacci sequence is a tool to quantify and understand patterns in nature.
Mathematics helps predict phenomena and make life easier.
The Fibonacci sequence is not just about numbers but also about understanding why things are the way they are.
Transcripts
so this will be the continuation of our
lecture the numbers in nature in terms
of fibonacci sequence
all we know that leonardo pisano
bergoglio he lived between 1170 and 1250
in italy
his nickname is fibonacci roughly means
son
of monachi he helped spread
the hindu arabic numerals through europe
in place of roman numerals he developed
the famous fibonacci sequence it
which means in one one two three
five eight thirteen twenty one thirty
four and thirteen fifty five
after learning this fibonacci sequence
you will appreciate
the mathematics and then you will love
it okay
so next slide so all we know this is 0
1 1 2 3 5 8 which
0 represents the a sub o 1 if sub 1
is up to a sub 3 so and so on
and so forth so starting with 0 and 1
each term is the sum of the two previous
terms
which is a 0 or a
three is equal to a plus one
there is a one plus one is equal to
further we are going to discuss that
okay next slide
so according to fibonacci one is
probably the most famous number sequence
it is named after the italian
mathematician leonardo
pisano of pisa known as fibonacci
he spoke is library apache
which introduced the sequence of to
western european mathematics although
the sequence has been described
earlier in indian mathematics
fibonacci sunflower seed pattern
soya sunflowers have 21 and 34 spirals
so have 85 89 89 and 44 depending on the
species
so pero however this pattern is not true
for all sunflowers
using six five seven sunflowers
according to swinton
found out that the one in five flowers
did not conform to the sweden
fibonacci sequence so we're gonna
contestant
so these are the example of the
fibonacci
in nature in flowers so we have the
mariposa or
lily guava mela
marigold even the banana
so these are the sequence of fibonacci
so 0 1 or 0 plus one is one
one plus one is two one plus two is
three
two plus three is five and three plus
five and eight
oh no i hoped
you're going to add this one 0 plus 1 is
1
1 plus 1 is 2 1 plus 2 is 3
and 3 plus 5 is 8. eight cyan i squared
the magnetic
field zero times zero is
zero one times one is one one times one
is
one two times two and so on sorry so an
e squared down
at the n so to check for that
so one plus one is two one plus four
is five nine nine four plus nine is
thirteen
and then nine plus twenty five is
straight thirty-four so now
sequence
so
fibonacci sequence one plus one plus
four is
six one plus one plus four four last
nine is fifteen
one plus one plus four plus nine plus
twenty-five is forty one plus one plus
four plus nine
plus twenty-five plus 64 is 104
sir
yeah but low 2 times
3 is 6. 3 times 5 is 15.
5 times 8 is 40 and 8 times 13
is 104.
it on 3 5 8 13 are
fibonacci sequence even the two three
five eight
two three five eight three five eight
thirteen so and i'm getting that jan
all right let's try to square this one
zero squared plus one squared
plus one squared plus two squared plus
three squared plus five squared
plus 8 squared plus 13 squared is equal
to 8 by
13 which is 104 they are the same diva
and do not forget that okay
so this will be the fibonacci squares
zero plus one is one one plus one
is two about one plus two
is three and then two plus three is five
so three plus five is eight
so negative nothing square network is
eight by eight
this is pi by pi three by three two by
two and one by one so
what is the area of rectangle
height is eight but nothing is five
plus eight that is thirteen so we have
and eight by 13 which is 104
so you know nothing squares
rate of rabbits
fibonacci created the problem that
concerning the birth of
rape of rabbits so at the beginning of a
month you are given a pair of new
rabbits so after a month the rabbit have
produced
no offspring however every month
thereafter so the pair of rabbits
produce another
pair of rabbits so the offsprings
produce in exactly the same manner
or in none of the rabbits dies how many
pairs of rabbits will there be at the
start of
succeeding months so the last of
fibonacci then discovered that the
number of pairs
pairs of rabbits for any month after the
first two
months can be determined by adding the
numbers of pairs of rabbits
in each of the two previous mods
for example
after a month
one plus one is two one plus two is
three and then two plus three
is five three plus five is eight so
union
the number of pairs of rabbits at the
start of each month is one
one two three five so and and so forth
so union
is a fibonacci sequence
opening mandolin golden ratio so
all we know that golden ratio exists
when a
line is divided into two parts and the
ratio of the longer
part a to the shorter part b is equal to
the
relation of the sum of a plus b to b
so finally n
plus b over a plus b so we have the
by all coordinates so which is value
1.618034
so again three divide two divided by
three eight divided by
so you know but you need more any two
successive numbers in the formula g
sequence
one one two three five eight and three
and so on
so very closely as a body and golden
ratio diva
your five over three one point six six
seven
and then eight by three one point six
thirteen by eight is one point six
two five zero so twenty one up with
thirteen so one point six
one by four so apache negative
so so these are the example of
fibonacci wait for the previous so this
is the formula
of the ratio quantity of one plus square
root of five
all over by two so we have one point six
one eight oh four
eight zero three four i mean so that
will be the value of the
golden ratio so these are the example of
fibonacci
sequence so it is
the golden ratio yeah
so these are the spirals examples
sunflower
even your cactus and or even this
sleeve of snails and other
so these are the examples spirals so
many plants grow in spirals
often the number of spirals is a
fibonacci numbers and the
spiral resemblance the fibonacci spiral
and
so what are the generalizations can we
say about mathematics chambering
many patterns and occurrences exist in
nature
in our world in our life
mathematics helps make sense of these
patterns and occurrences
and all we know that mathematics is a
tool to quantify
organize and control the world predict
phenomena
and make life easier easier for us
mathematics is not just solving for your
ex
or looking for your ex it's also
figuring out why
so you wanna ex
so this already generalization
about mathematics so
zero one one plus one one plus two
two plus three three plus five so
i hope you learned a lot in the
fibonacci sequence
so enjoy your enjoy our lecture and see
you
another lecture bye
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