P2- Fibonacci Sequence | Golden Ratio | Patterns and Numbers in Nature | Math1/GE3

Summary

In this educational video, Teacher Juice explores patterns and numbers found in nature, illustrating how they can be mathematically represented. From snail shells to honeycombs, patterns like spirals, tessellations, and symmetry are discussed. The Fibonacci sequence, with its prevalence in nature and art, is highlighted, along with the Golden Ratio, which is approximately 1.618. The video connects these mathematical concepts to their presence in architecture and nature, emphasizing their significance in our world.

Takeaways

• 🔍 Patterns in nature are regular, repeated forms or designs that help us organize information and understand the world around us.
• 🐌 Examples of natural patterns include the spirals found in snail shells, pine cones, and the Milky Way galaxy.
• 🧩 Tessellations are patterns where shapes fit perfectly together without overlaps or gaps, like honeycombs or floor tiles.
• 🦓 Spots and stripes on animals are patterns that result from reaction-diffusion processes.
• 🪴 Symmetry, such as reflection or mirror symmetry, is a pattern where one part of an object is the mirror image of another.
• 🌼 The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1.
• 🌺 The Fibonacci sequence is often observed in the number of petals in flowers and other natural phenomena.
• 🔢 The golden ratio, approximately 1.618, is derived from the Fibonacci sequence and is found in various aspects of art, architecture, and nature.
• 🏛 Famous structures like the Parthenon, Taj Mahal, and Egyptian pyramids incorporate the golden ratio in their design.
• 🌈 The Fibonacci spiral, created by connecting squares of Fibonacci numbers, is a visual representation of the sequence and is found in various natural forms.

Q & A

What is a pattern according to the video?

A pattern is defined as a regular, repeated, or recurring form or design, especially in the natural world.

Why are patterns in nature important?

Patterns in nature help us organize information and make sense of the world around us.

What are the steps to follow to understand math in the context of patterns?

The steps are: 1) to find a pattern, 2) represent patterns in the form of symbols, notations, shapes, or numbers, and 3) interpret the pattern.

What is an example of a pattern in the video?

An example given is the blue lines progressively getting thicker until they would take over the whole square.

How are missing terms in a sequence determined in the video?

The missing terms are determined by identifying the pattern in the sequence, such as adding 4 to the preceding number to find the next term.

What are the different types of patterns found in nature mentioned in the video?

The types of patterns include spirals, tessellations, spots and stripes, and symmetry.

Can you explain what a spiral pattern is as described in the video?

A spiral pattern is curved and starts from a small point, moving farther away as it revolves, getting bigger but maintaining the same pattern.

What is tessellation and how is it related to patterns in nature?

Tessellation is a pattern of shapes that fit perfectly together without overlaps or gaps, like honeycombs or floor tiles.

How are spots and stripes patterns formed in animals?

Spots and stripes patterns in animals are the result of a reaction-diffusion process.

What is symmetry and how does it relate to patterns?

Symmetry means that one shape becomes exactly like another shape when moved in some ways, indicating that parts of an object are mirror images of each other.

What is the Fibonacci sequence and how is it related to patterns in nature?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. It is often found in patterns in nature, such as the number of petals in a flower.

How is the golden ratio derived from the Fibonacci sequence?

The golden ratio is derived by dividing two consecutive Fibonacci numbers, and it is approximately equal to 1.618.

Why is the golden ratio significant in art, architecture, and design?

The golden ratio is significant because it is considered aesthetically pleasing and is found in many famous architectural structures and works of art.

Outlines

00:00

🌿 Patterns in Nature and Mathematics

This paragraph introduces the concept of patterns in nature, defined as regular, repeated, or recurring forms or designs. Examples such as snail shells, honeycombs, and pine cones are given to illustrate natural patterns. The paragraph emphasizes the importance of patterns in organizing information and understanding the world. It outlines the steps to understand math through patterns: finding a pattern, representing it with symbols or numbers, and interpreting it. The video provides examples of pattern recognition, such as the progression of blue lines in a square and finding missing terms in a numerical sequence. The examples demonstrate how patterns can be identified and used to predict missing elements.

05:03

🔢 Fibonacci Sequence and Golden Ratio

The second paragraph delves into the Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones, often starting with 0 and 1. The sequence is highlighted by the fact that it appears in various aspects of nature, such as the number of petals in flowers. The paragraph explains the historical significance of the Fibonacci sequence, attributing its popularization in Europe to Leonardo Fibonacci. It also describes how to calculate subsequent terms in the sequence and provides an example of calculating the ninth term. The concept of the golden ratio, approximately 1.618, is introduced as a result of dividing two consecutive Fibonacci numbers. The golden ratio's influence on art, architecture, and design is discussed, with examples like the Parthenon and the Taj Mahal.

10:05

🎨 Golden Ratio in Art and Architecture

The final paragraph of the script discusses the prevalence of the golden ratio in human creations, suggesting that this mathematical pattern is not just a coincidence but a fundamental aspect of the world we live in. It mentions how the golden ratio has inspired many to incorporate it into their works, leading to aesthetically pleasing and harmonious designs. The paragraph also touches on the philosophical and aesthetic implications of the golden ratio, hinting at its role in helping us understand the beauty and order in nature and human-made structures.

Keywords

💡Pattern

A pattern refers to a regular, repeated, or recurring form or design. In the context of the video, patterns in nature are visible regular forms found in the natural world, such as the spirals in a snail shell or the hexagonal cells in a honeycomb. The video emphasizes how recognizing patterns helps us organize information and make sense of the world around us, highlighting the importance of patterns in understanding both natural and mathematical phenomena.

💡Spiral

A spiral is a curved pattern that starts from a small point and moves outward, getting larger as it revolves. The video uses examples like the snail shell and the pine cone to illustrate spirals in nature. Spirals are significant in the video as they represent one of the fundamental patterns that can be mathematically modeled and are prevalent in various forms and contexts.

💡Tessellation

Tessellation is a pattern where shapes fit perfectly together without overlaps or gaps. The video mentions honeycombs and floor tiles as examples of tessellation. Tessellations are important in the video's theme as they demonstrate how patterns can be used to fill space efficiently and are a common occurrence in both natural and man-made structures.

💡Spots and Stripes

Spots and stripes refer to patterns commonly found in animals, resulting from reaction-diffusion processes. The video suggests that these patterns are not random but are part of the natural world's order. They are used in the video to show how patterns can be observed in living organisms, contributing to the overall theme of recognizing order in nature.

💡Symmetry

Symmetry in the video is defined as a property where one shape becomes exactly like another when moved in certain ways. It is illustrated with examples such as leaves, apples, and butterflies, where one part is a mirror image of the other. Symmetry is a key concept in the video, as it highlights the aesthetic and functional aspects of patterns in nature and mathematics.

💡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. The video explains this sequence through the context of the growth of a rabbit population and its prevalence in nature, such as the number of petals in a flower. The Fibonacci sequence is central to the video's message, showing how mathematical patterns are inherent in natural phenomena.

💡Golden Ratio

The golden ratio, approximately equal to 1.618 and represented by the Greek letter phi (φ), is a mathematical concept that describes a special kind of proportion. The video explains that the golden ratio can be derived from consecutive Fibonacci numbers and is found in various aspects of art, architecture, and nature, such as the Parthenon and the Taj Mahal. It is a key concept in the video, illustrating the harmony and balance found in patterns.

💡Reflection Symmetry

Reflection symmetry, also known as mirror symmetry, is a type of symmetry where a shape can be divided by an imaginary line, and the two halves are mirror images of each other. The video uses examples like leaves and butterflies to illustrate reflection symmetry. This concept is integral to the video's exploration of patterns, emphasizing the recurring themes of balance and order in nature.

💡Translation Symmetry

Translation symmetry is a type of symmetry where a pattern is repeated at regular intervals in a specific direction. While not explicitly detailed in the video script, it is implied in the discussion of patterns that repeat in nature and can be observed in various natural and man-made structures. This concept supports the video's overarching theme of identifying and understanding patterns.

💡Radial Symmetry

Radial symmetry is a type of symmetry where an object can be divided into symmetrical halves by any number of planes passing through a central point. The video does not provide specific examples, but radial symmetry is a concept that complements the discussion of patterns in nature, such as the arrangement of petals in a flower or the segments of a sea star.

💡Mathematical Modeling

Mathematical modeling in the video refers to the process of creating mathematical representations of real-world phenomena. The video discusses how patterns found in nature, such as spirals and tessellations, can be mathematically modeled to understand and predict their behavior. This concept is crucial to the video's theme, as it bridges the gap between the natural world and the abstract world of mathematics.

Highlights

Patterns are regular repeated or recurring forms or design visible in the natural world.

Examples of patterns in nature include snail shells, honeycombs, and pine cones.

Patterns help us organize information and make sense of the world around us.

Steps to understand math include finding a pattern, representing it, and interpreting it.

An example pattern involves blue lines progressively getting thicker.

Finding missing terms in a sequence involves identifying the pattern of addition.

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

The Fibonacci sequence starts with 0 or 1 and is found in various natural phenomena.

The golden ratio, approximately 1.618, is derived from consecutive Fibonacci numbers.

The golden ratio is used in architecture and art to create aesthetically pleasing proportions.

Spiral patterns, such as those found in snail shells and galaxies, are a common form in nature.

Tessellations are patterns where shapes fit perfectly together without overlaps or gaps.

Spots and stripes patterns in animals are the result of reaction-diffusion processes.

Symmetry, including reflection, translation, and radial symmetry, is a pattern found in nature.

Mathematics is present in nature, and patterns like the Fibonacci sequence are translated into mathematical symbols.

The Fibonacci spiral can be visualized using square tiles, revealing a pattern seen in nature and architecture.

Many famous architectural structures incorporate the golden ratio in their design.

Patterns in nature help us unravel the mysteries of the world we live in.

Transcripts

hi everyone this is teacher juice once

again and today's video we're going to

talk about lessons 1.2 to 1.3 which is

the patterns and numbers in nature and

the world now let us try to define first

what is pattern

patterns are regular repeated or

recurring forms or design

patterns in nature are visible regular

forms found in the natural world

so example we have here the snail shell

the honeycomb and the pine cone

these are examples of patterns in nature

this helps us organize informations and

make sense of the world around us if you

can still remember the steps to follow

to understand math which is the first

to find a pattern

second

represent patterns in the form of

symbols notations shapes or numbers and

finally

interpret the pattern

now let us have this example and use the

steps to answer the following

which of the given shapes would complete

the pattern

what do you think it's letter

a

the blue lines are progressively getting

thicker to the point where the lines

would take over the whole square now let

us have example number two

find the missing terms in the sequence

8

blank

16 blank

24 28 and 32.

the answer is

12 and 20 very good

three consecutive numbers 24 28 and 32

are examined to find the sequence

pattern

and these are the rules obtained

you can notice that the corresponding

number is obtained by adding 4 to the

preceding number

therefore the missing terms are 8 plus 4

is 12 and 16 plus 4 is 20. that is why

our answer is 12 and 20.

[Music]

millions of patterns can be found in the

environment

these patterns occur in various forms

and in different contexts

which can be modeled mathematically

first kind of pattern is the spiral

spirals are curved which starts from a

small point

moving farther away as it revolves

around it gets bigger and bigger but

the pattern is not changing

we have your example the snail shell

the pine cone and even our galaxy the

milky way these are examples of

the spirals

next kind are tessellations tessellation

is a pattern of shape that fits

perfectly together and having no

overlaps and gaps for example we have

here the honeycomb or even the our floor

tiles these are examples

of tessellation next kind are the spots

and stripes

patterns like stripes and spots

are commonly present in different

organisms especially in animals

these spots and stripes are result of a

reaction diffusion

so you can see there

the animals have their own spots and

stripes

and we have here

symmetry

symmetry comes from a greek word which

means to measure together

mathematically

it means that one shape becomes exactly

like another shape

when you move it in some ways

it indicates that you can draw an

imaginary line across an object and the

resulting part are mirror image of each

other so for example we have here the

leaf

the apple and the butterfly

this demonstrates symmetry because the

left part of it is a mirror image of the

right part

and that is the first kind of symmetry

which is the reflection or the mirror

symmetry next is the translation

symmetry and the last kind is the radial

symmetry

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 numbers of

petals in a flower

you might think 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 seems a coincidence to

you but this sort of numbers form a

pattern

in the sequence the next number is found

by adding up the two numbers before it

for example

it begins with zero

next is one to find a third number we

add last two numbers

zero plus one equals one

continuing on one plus one equals two

to get the next number we will add the

previous two again one plus two equals

three two plus three is five three plus

five is eight

we will then get 13 21 34 55 89 and so

on

this pattern is called the fibonacci

sequence

this pattern was popularized in europe

by leonardo of pisa

also known as leonardo fibonacci thus

the name of the pattern originated

he is one of the most influential

mathematicians of the middle ages

because hindu arabic numeral system

which we also we use today was

popularized in the western world by

leonardo fibonacci

in his book the book of calculation

fibonacci post and solve a problem

involving the growth of a population of

rabbits based on idealized assumption

the solution for this problem

the fibonacci sequence

fibonacci sequence is a wonderful series

of numbers that could either start with

0 or 1. now let us try to determine the

next few terms the first term is the f

of 1 or the 1.

the second term the f of two is still

the same one

the third term or the f of three is two

f of four is three

f of five is five

f of six is eight

f of seven or the 7th term is 13 and so

on

now what if we are going to look for the

f of n or the n term in the fibonacci

sequence

we need to add

f sub n minus 1 plus

f of n minus 2 where f of n minus 1 is

the previous term and f of n minus 2 is

the previous previous term

given the series let us try to find the

ninth term of the fibonacci sequence

in the given we only have until f of

seven or the seven terms so let us try

to solve first for the f of eight

we have here add the previous plus the

previous previous term we have 8 plus 13

is 21 and we are now ready

to get or to find the 9th term which is

13 plus 21 is equal to 32 so therefore

the ninth term of the fibonacci sequence

is

32.

now let us have this another example

what is f of three plus f of seven minus

f of six

so first let us try to substitute first

the given terms

f of three is

two

f of 7 is

13

and f of 6 is

8. so 2 plus 13 minus 8 is

7. therefore the answer is seven

let's visualize these numbers using a

square tiles let's start with a one by

one square

then another

together they form a one by two

rectangle

above is a two by two square

next to it

is a three by three square

beneath is a five by five square

and if we continue to do this

and connect opposing diagonals

continuously it will reveal the

fibonacci spiral

[Music]

and this spiral could be seen a lot in

nature architecture

arts human body and beyond

going back to the rectangle what if we

are going to divide the two dimensions

8 and 13 in this case notice that it is

just 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

v or phi

in mathematics the golden ratio is used

to describe the relationship of the two

figures where the numbers seem to be in

a form of a 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 are

somehow near

1.618

since the discovery of this golden ratio

many known individuals were inspired to

incorporate this magnificent number to

the great works and creations

man's marvelous architectures like

parthenon

taj mahal

roman arches

egyptian pyramids

eiffel tower and many more were also

built based on this mathematical pattern

[Music]

coincidence or not

this pattern become part of the world we

live in

they help us unravel the mystery of

nature

[Music]