1. Fibonacci Series, Pascal’s Triangle, and The Golden Ratio
“Where there is matter, there is geometry.”
~Johannes Kepler

2. Fibonacci Sequence (spiral) in Nature

3. Fibonacci Sequence (spiral) in Nature

4. Fibonacci Sequence in Nature

5. Fibonacci Sequence (spiral) in Nature

6. Fibonacci Series 1, 1, 2, 3, 5, 8, 13, 21, 34… Fibonacci Series
Begin with one male rabbit and female rabbit that have just been born.
Rabbits reach maturity after one month.
The gestation period of a rabbit is one month.
After reaching maturity, female rabbits give birth every month.
A female rabbit gives birth to one male rabbit and one female rabbit.
Rabbits do not die.
Start at 0 month with newborn male and female
End of the first month = 1 pair
End of the second month = 2 pair
End of the third month = 3 pair
End of the fourth month = 5 pair
243 pairs of rabbits produced in one year
1, 1, 2, 3, 5, 8, 13, 21, 34… Fibonacci Series
The next number is found by adding up the two numbers before it.
The 2 is found by adding the two #’s before it (1+1)
The 3 is found by adding two #’s before it (1+2)…….
Fibonacci Series

7. Creating Pascal’s Triangle
At the top of Pascal’s Triangle is the number 1 which makes up the zero-th row.
The first row contains two 1’s, both formed by adding the two numbers above them to the left and right, in this case (all #’s outside triangle are 0’s).
Do the same to create 2nd row. 1+0=1, 1+1 =2, 1+0 =1
3rd row: 1+0=1, 1+2=3, 2+1=3, 1+0=1
4th row: 1+0=1, 1+3=4, 3+3=6, 3+1=4, 1+0=1
Row: 1 1 2 3 4 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 5

8. Fibonacci Numbers found in Pascal’s Triangle
1+0=1
1+0=1
1+1=2
2+1=3
1+3+1=5
1+4+3=8
=13
Fibonacci Numbers found in Pascal’s Triangle

9. Fibonacci Series from Pascal’s Triangle

10. Fibonacci Sequence (spiral) in Nature 1, 1, 2, 3, 5, 8, 13, 21, 34….

11. Pascal’s Triangle This is an example of a fractal.
All the odd numbers are in black, and all of the even numbers are in white.
Pascal’s Triangle

12. A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.
Fractals

13. Fractal, Fractal, Fractal, Fractal, Fractal, fractal

14. Fractal Art

15. Creating Fibonacci’s Spiral1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8 5 8 1 1 2 3
Fibonacci Sequence
Fibonacci Spiral

16. Fibonacci Sequence and Spiral

17. The Golden Ratio a/b = 8/5 = 1.6 8 + 5 = 1.625 8 a/b = 5/3 = 1.67 or
8 + 5 = 8
a/b = 5/3 = 1.67 or
5 + 3 = 1.6 5 3 5 8
The Golden Ratio

19. Golden Ratio ( - phi) in Fibonacci Sequence
The ratio of consecutive numbers in the Fibonacci sequence.
Golden Ratio ( - phi) in Fibonacci Sequence
The Mind-Blowing Mathematics of Sunflowers

20. Fibonacci Sequence in Nature – the Golden Ratio

23. The Golden Ratio in the Human Body

24. The Golden Ratio In Human Body

25. Fibonacci Spiral

26. Golden Ratio in Architecture