The Golden Ratio is a quantity roughly equal to 1.618 This value is represented by the symbol phi . The Golden Ratio is ls known as the Golden Section, Golden Mean, Extreme and Mean Ratio. Medial Section, Divine Proportion, Divine Section, Golden Proportion, Golden Cut, and Golden Number. This is shown algebraically by the equation below.

Phi and phi

Phi and phi are, just like many other names for irrational numbers, greek letters. They are both associated with the golden ratio we showed in the last slide.

Phi and phi are conjugates of one another. Here are their equations shown below. Capital Phi is equal to the Golden Ratio taken in reverse.

The Golden Spiral

The Golden Ratio with the Fibonacci numbers

The Fibonacci numbers are directly related to the Golden Ratio. If you take a term in the Fibonacci Sequence and divide it by the term before it, you will get a number approaching phi (1.618...). This is shown by the below equation.

The Golden Rectangle

Fibonacci in Nature

Fibonacci breeding

patterns

In nature, the Fibonacci number does not just

show up in flowers and spirals, but in breeding patterns of animals, and tree branches. Look

at this graph. Each pair needs one month to mature, and after that it reproduces a new pair each month. If you look at how many rabbit pairs there are each month, you will see the Fibonacci pattern. Same

with tree branches. Each new branch needs time

to grow and be strong enough to support a

new branch, giving us the same pattern

with the rabbits.

Fibonacci numbers

The Fibonacci Concept

The Fibonacci series was created by mathematician Leonardo of Pisa also known as Filius Bonacci or Fibonacci. The concept was simple. You start with 1 and add 1 to get 2. Then you take that number and add it to the previous one to get the next number in the sequence. This sequence is unlike most sequences because each number is defined recursively, meaning that it is in terms of the previous number. It was a simple concept, but it turned out to give us some amazing answers to how nature is built, along with our bodies and the universe.

**Fibonacci Numbers**

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89....

**Made by Bryce, Gordon,**

and Rodrigo

and Rodrigo

**Thanks for watching!**

Although the Fibonacci sequence had been discovered in India before, we credit Fibonacci, with their discovery; he mentioned them in his book, Liber Abaci 2.

Fibonacci proposed a mathematical problem in which a pair of rabbits bred each month to yield a new pair of rabbits, and asked the question of how many rabbits there would be in a year? In his theoretical problem, no rabbits died and it would take exactly one month for the rabbits to grow old enough to mate.

Fibonacci found that the number of rabbits in a month was equal to the sum of the number of rabbits in the two previous months. This can be described as the following: F_n=F_(n-1)+F_(n-2)

-Every 3rd term in the Fibonacci sequence is a multiple of 2

-Every 4th term is a multiple of 3

-Every 5th term is a multiple of 5.

So that:

2 = 1 + 1

3 = 2 +1

5 = 3 + 2

8 = 5 + 3

ArithmeticPatterns

2, 3, 5

2 * 5 = 10

3 * 3 = 9

10 - 9 = 1

A common misconception is that the Fibonacci numbers are equal to the Lucas numbers. However, this is not true, the Lucas numbers start with 1 and 2 while Fibonacci numbers start with 1 and 1.

Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

Lucas Numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76...

Common Misconceptions of Fibonacci Numbers

1, 1,

2

,

3

,

5

,

8

, 13,

21

,

34

,

55

, 89....

-The sum of two Fibonacci numbers squared results in another Fibonacci number.

1

^2 +

1

^2 =

2

1

^2 +

2

^2 =

5

2

^2 +

3

^2 =

13

Many people know about Fibonacci numbers in nature, but do not know why. Next time you find a pine cone, count the spirals on it. You will probably find that it will have a number of spirals that is a Fibonacci number. That is just one amazing thing though. Remember the golden ratio? That ratio shows up in how leaves are spaced. If the degrees between the leaves were 360/an integer, the leaves would miss sunlight, but the golden ratio insures that no two leaves will ever overlap. The most amazing thing about the leaves is that it happens in plants all on its own

Fibonacci Numbers Expressed Geometrically

A series of squares whose sides correspond to a series of Fibonacci numbers and arrange them in a outwardly spiral pattern.

1

^2 +

1

^2 +

2

^2 +

3

^2 +

5

^2 +

8

^2= 104 =

8

*

13

Series: 1+2+3+4

Sequence: 1,2,3,4

Petals

If you have ever stopped and

smelled the roses, you probably have not tried to count the number of petals on it. If you count the number of petals on just about any flower, you will come up with an amazing phenomenon. Many plants have petals equal to a fibonacci number. such as daisies sometimes even have 43!

1 petal: lily

2 petals: euphorbia

3 petals: trillium

5 petals: columbine

8 petals: bloodroot

13 petals: Black eyed susan

It is also worthwhile to mention that we have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.