FIBONACCI NUMBERS AND THE GOLDEN SPIRAL Leonardo Pisano Bigollo - - PowerPoint PPT Presentation
FIBONACCI NUMBERS AND THE GOLDEN SPIRAL Leonardo Pisano Bigollo (Fibonacci) (c. 1170 c. 1250) Italian Mathematician Considered by some the most talented western mathematician of the Middle Ages Published Liber Abaci (Book of
Leonardo Pisano Bigollo (Fibonacci)
(c. 1170 – c. 1250) Italian Mathematician Considered by some “the most talented western mathematician
Published Liber Abaci (Book of Calculation) in 1202 in which he introduced western mathematics to a mathematic sequence known today as the Fibonacci numbers
Fibonacci did not discover the series but he introduced it to the west in his 1202 book Liber Abaci (Book of Calculation). In his book Fibonacci considered the growth of an unrealistic rabbit population. A newly born rabbit pair, one male and one female, was put into a field. After one month they were able to mate and one month later the female gave birth to another pair of rabbits, a male and a female. The assumption of the idealized rabbit population were that rabbits never die, a rabbit pair could mate one month after birth and would reproduce another pair of rabbits, a male and a female, one month later. Fibonacci wanted to know how many rabbits there would be after one year. At the end of the second month the original female produces another pair making a total of two pairs. At the end of the third month the
the end of the fourth month the original female produces another pair and her first pair reproduces their first pair making a total of five pairs. So how many rabbits will there be at the end of one year?
Pairs ready Reproduction # # to procreate Process Pairs Rabbits Now None 1 2 1 1 1+1= 2 4 2 1 2+1= 3 6 3 2 3+2= 5 10 4 3 5+3= 8 16 5 5 8+5= 13 26 6 8 13+8= 21 42 7 13 21+13= 34 68 8 21 34+21= 55 110 9 34 55+34= 89 178 10 55 89+55= 144 288 11 89 144+89= 233 466 12 144 233+144= 377 754 Month
𝒇𝒚 𝒇𝒚 A single grain of rice doubled every month R1 earning 500% interest continuously compounded Amount of rabbits after 1 year Pairs of rabbits after 1 year
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765… The sequence starts with 0 and 1, and each subsequent number is the sum of the previous two. Mathematically defined as: Fn = Fn-1 + Fn-2 with seed values F0 = 0, F1 = 1 Squares with sides equal to the Fibonacci numbers place next to each
equal to the sides of the squares connecting the opposite corners of the squares form a spiral known as the golden spiral
1 1 1 1 1
2
1 1
3 2
1 1
3 2 5
1 1
3 2 8 5
1 1
13 3 2 8 5
1 1
21 13 3 2 8 5
1 1
21 13 3 2 8 5 34
1 1
21 13 55 3 2 8 5 34
1 1
89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
The Fibonacci numbers is closely related to the golden ratio of 1.6180 or its inverse of 0.618. The ratio of each Fibonacci number to the previous Fibonacci number resembles the golden ratio; the higher up the sequence, the more closely the ratio of two consecutive Fibonacci numbers will approach the golden ratio. The Fibonacci number sequence and the golden ratio can be used to describe the proportions of almost everything in nature
from the smallest DNA sequence to celestial
bodies.
Fibonacci nr (Fn) 1 2 1 3 1 1.0000 1.0000 4 2 2.0000 0.5000 5 3 1.5000 0.6667 6 5 1.6667 0.6000 7 8 1.6000 0.6250 8 13 1.6250 0.6154 9 21 1.6154 0.6190 10 34 1.6190 0.6176 … … … … 51 12586269025 1.6180 0.6180 52 20365011074 1.6180 0.6180 53 32951280099 1.6180 0.6180 54 53316291173 1.6180 0.6180 55 86267571272 1.6180 0.6180 Fn/Fn-1 1/(Fn/Fn-1) n
1 1
144 89 21 13 55 3 2 8 5 34
1 1
144 89 21 13 55 3 2 8 5 34
Financial markets also seem to conform to the Fibonacci sequence and the golden ratio. Technical analysts have developed some strategies to take advantage of this phenomenon. The golden ratio can be translated into the following percentages: 50%, 61.8% and 38.2% (100% - 61.8%). Additional multiples can also be added. These percentages along with the Fibonacci numbers can be used in technical strategies. The golden ratio and the Fibonacci sequence have been used with numerous successes to analyse and predict stock market moves.
Low Support Resistance High Golden Ratio %
FIBONACCI RETRACEMENTS
Low High Resistance Support 38.2% Arc 50% Arc 61.8% Arc
FIBONACCI ARCS
Low High 100% 0% 61.8% 50% 38.2% Resistance Support
FIBONACCI FANS
89 55 34 21 13 8 5 1 1 2 3
FIBONACCI TIME ZONES
1 1
144 89 21 13 55 3 2 8 5 34