1. Fibonacci Sequences and the Golden Ratio Carl Wozniak Northern Michigan University

2. Who was Fibonacci? Leonardo da Pisa (1170- 1240) – First true mathematician since the Greeks – Liber Abbaci (Book of Calculation, 1202) introduced the nine numerals and the concept of zero to Europe

3. Who was Fibonacci? In the same book Fibonacci presented a word problem concerning breeding rabbits.

4. The rabbit problem “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?”

5. The rabbit problem So, in five months we have: 1 2 3 5 8 pairs

6. We can continue the sequence 1 2 3 5 8 13 21 34 55 89 144 … Notice that each number is equal to the sum of the previous two numbers. This is the Fibonacci Sequence. The really neat thing is that we find these numbers in many places in nature.

7. Fibonacci numbers in nature Flower petals – lilies and iris have 3 petals; buttercups have 5 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.

8. Fibonacci numbers in nature Seed heads – Counting along the spirals of seed heads normally leads to a Fibonacci number.

9. Fibonacci numbers in nature Pine cones – Pine cone scales are also normally arranged in a Fibonacci spiral

10. Fibonacci numbers in nature We also find Fibonacci numbers in: – The scales of a pineapple – The number of leaves around the circle of the stem – The number of leaves until another leaf is directly above the leaf where we started counting – About 90% of all plants exhibit some form of Fibonacci sequencing

11. So why are they there? What are your thoughts?

12. So why are they there? Reasons others have given – In the case of plants, the arrangement maximizes the exposed area of each leaf – Provides maximal surface to continue growth (coiled shell growth)

13. So why are they there? Reasons others have given – Maximizes space in seed heads

14. The Golden Ratio Divide each number in the Fibonacci sequence by the number immediately preceding it. 1 / 1 = 1, 2 / 1 = 2, 3 / 2 = 1.5, 5 / 3 = 1.666..., 8 / 5 = 1.6, 13 / 8 = 1.625, 21 / 13 = 1.61538... You find that you get closer and closer to the number 1.618…. The ratio of 1.618:1 is the Golden Ratio, and it is also frequently found, not only in nature, but in human constructions.

15. The Golden Ratio The Golden Ratio is also known as the Golden Mean, Golden Section and Divine Proportion. It is a ratio or proportion defined by the number Phi ( = 1.618033988749895... ) In the following illustration, A is to B as B is to C. This occurs only where A is 1.618... times B and B is 1.618... times C.

16. Aesthetics Which of the following rectangles do you find most appealing?

17. Aesthetics Well, how about this grouping?

18. The Golden Ratio in art In daVinci’s The Last Supper In the front view of the Acropolis In the construction of a violin

19. The Golden Ratio in nature In a nautilus shell, each subsequent chamber is approximately 1.68 times larger than the last.

20. The Golden Ratio in you You can find a number of instances in your own body that approximate phi daVinci’s Vitruvian Man

21. The Golden Ratio in you The lengths of your finger joints The distance from the floor to your navel relative to your height Front two incisors height to width Ratio of forearm and hand length