Fibonacci Sequence

Summary

This 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

00:00

📐 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.

05:06

🌱 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.

10:08

🔢 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.

Keywords

💡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. This sequence is a key concept in the video, as it is central to the discussion of mathematical patterns found in nature. The script mentions that Fibonacci numbers appear in various natural phenomena, such as the spiral patterns in sunflowers and pine cones.

💡Leonardo Pisano (Fibonacci)

Leonardo Pisano, also known as Fibonacci, was an Italian mathematician who lived between 1170 and 1250. He is credited with popularizing the Hindu-Arabic numeral system in Europe and is best known for his development of the Fibonacci sequence. The video script discusses his contributions to mathematics and how he helped to replace Roman numerals with the more efficient Hindu-Arabic numerals.

💡Hindu-Arabic Numerals

Hindu-Arabic numerals refer to the decimal number system that is used worldwide today. In the video, it is mentioned that Fibonacci played a significant role in spreading this numeral system throughout Europe, which replaced the less convenient Roman numerals. This development was pivotal in advancing mathematical and scientific computation.

💡Golden Ratio

The Golden Ratio is a mathematical constant that is found by dividing a line into two parts such that the ratio of the whole length to the longer part is the same as the ratio of the longer part to the shorter part. The value of the Golden Ratio is approximately 1.618. In the script, it is mentioned that the Golden Ratio is closely related to the Fibonacci sequence, as the ratio of successive Fibonacci numbers approximates the Golden Ratio.

💡Spiral Patterns

Spiral patterns are a common occurrence in nature, often following the Fibonacci sequence. The video script gives examples of sunflowers, pine cones, and even snail shells exhibiting spiral patterns that correspond to Fibonacci numbers. These patterns are not only aesthetically pleasing but also serve practical purposes in plant growth and seed distribution.

💡Sunflowers

Sunflowers are used in the script as a prime example of Fibonacci numbers in nature. They often have spiral patterns with the number of spirals corresponding to Fibonacci numbers, such as 21 and 34. However, the script also notes that not all sunflowers conform to this pattern, indicating that while Fibonacci numbers are prevalent, they are not universal.

💡Mariposa Lily

The Mariposa Lily is another example of Fibonacci numbers in nature mentioned in the script. Like sunflowers, it exhibits patterns that follow the Fibonacci sequence, illustrating how these mathematical principles are embedded in the structure of various flora.

💡Mathematical Patterns

Mathematical patterns are recurring rules or sequences that can be found in various aspects of life and the natural world. The video emphasizes how mathematics, including the Fibonacci sequence, helps us understand and quantify these patterns, showing the interconnectedness of math and nature.

💡Rabbit Problem

The Rabbit Problem, as mentioned in the script, is a classic example that Fibonacci used to illustrate the growth of the Fibonacci sequence. It involves a scenario where a pair of rabbits produces another pair each month, and the number of rabbit pairs grows according to the sequence. This problem is a fun and engaging way to demonstrate the practical application of the sequence.

💡Quantify

To quantify is to measure or express the magnitude of something in numerical terms. In the context of the video, quantifying refers to using mathematics to measure and understand patterns in nature. The script suggests that mathematics is a tool for quantifying, organizing, and predicting phenomena in the world around us.

💡Organize

Organizing, in the video's context, refers to the way mathematics helps structure and categorize information. By using mathematical concepts like the Fibonacci sequence, we can impose order on complex natural phenomena, making them more comprehensible and manageable.

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