The document discusses the Fibonacci sequence, which is a series of numbers where each number is the sum of the two preceding numbers. It describes how the ratio of numbers in the Fibonacci sequence approaches the golden ratio as the sequence progresses. Examples of the Fibonacci sequence appearing in nature are given, such as sunflowers, shells, waves, and galaxies exhibiting the sequence in their patterns. The document also covers how the Fibonacci sequence and golden ratio have been used in art and architecture.

1. The Nature of
Mathematics

2. The Fibonacci Sequence
 A sequence of numbers in which each
member is the sum of the two preceding
numbers. For example
0,1,1,2,3,5,8,13,21 and so forth.

3. The Fibonacci Sequence
 the fibonacci sequence was invented by
Leonardo Fibonacci

4.  The Golden Ratio
 The golden ratio is ϕ(phi) = 1.61803…..
 Taking the ratio of two successive numbers in
the sequence will approach the golden ratio
as you move deeper into the sequence
0,1,1,2,3,5,8,13,21,34,55…..
5/3= 1.67 8/5=1.6 13/8=1.625 34/21=
1.619

5. Fibonacci spiral

6. Fibonacci sequence in nature
 The Sunflower
 shell
 Flower petals
 Ocean wave
 Galaxy

7. Fibonacci in petals
FLOWERS NUMBEROF
PET
ALS
Lilies 3
Buttercups 5
Delphiniums 8
Marigolds 13
Asters 21
Daisies 34, 55, 89

8. Fibonacci sequence in nature

9. Fibonacci in art
 Leonardo da Vinci
Painted “Mona Lisa”
During renaissance
With divine proportion

10. Mathematics in universe
Constellation

11. Mathematics in universe
 Example of Constellation

12. Thank you !