P2- Fibonacci Sequence | Golden Ratio | Patterns and Numbers in Nature | Math1/GE3
Summary
TLDRIn 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
- π Patterns in nature are regular, repeated forms or designs that help us organize information and understand the world around us.
- π Examples of natural patterns include the spirals found in snail shells, pine cones, and the Milky Way galaxy.
- 𧩠Tessellations are patterns where shapes fit perfectly together without overlaps or gaps, like honeycombs or floor tiles.
- π¦ Spots and stripes on animals are patterns that result from reaction-diffusion processes.
- πͺ΄ Symmetry, such as reflection or mirror symmetry, is a pattern where one part of an object is the mirror image of another.
- πΌ The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1.
- πΊ The Fibonacci sequence is often observed in the number of petals in flowers and other natural phenomena.
- π’ The golden ratio, approximately 1.618, is derived from the Fibonacci sequence and is found in various aspects of art, architecture, and nature.
- π Famous structures like the Parthenon, Taj Mahal, and Egyptian pyramids incorporate the golden ratio in their design.
- π 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
πΏ 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.
π’ 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.
π¨ 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.
Mindmap
Keywords
π‘Pattern
π‘Spiral
π‘Tessellation
π‘Spots and Stripes
π‘Symmetry
π‘Fibonacci Sequence
π‘Golden Ratio
π‘Reflection Symmetry
π‘Translation Symmetry
π‘Radial Symmetry
π‘Mathematical Modeling
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]
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