Title: Fibonacci Numbers and The Golden ratio in nature Charlie Reid
1 Fibonacci Numbers and The Golden ratioin natureCharlie Reid 2 What do all these have in common They are all beautiful! 3 Mathematicians think they have discovered the maths behind it!
Explain Fibonacci numbers
How this relates to Phi (The Golden Ratio)
How this relates to human perception of beauty
Class Guinea pig!
4 Leonardo de Pisa filius Bonaccio Fibonacci (1170 1240)
Educated in North Africa due to father
Accompanied father on travels
Finished travelling in 1200 started work on Liber Abaci (Book of Calculating)
Tried to revive ancient mathematics
Led to discovery of one of the most significant sequences ever discovered
5 Fibonacci Sequence
Rabbit Problem (Frederic II Holy Roman Emperor)
A pair of adult rabbits produce a pair of baby rabbits once every month.
Each pair of baby rabbits require one month to grow to be adults and subsequently produce one pair of baby rabbits each month thereafter.
Determine the number of pairs of adult and baby rabbits after some number of months.
It is assumed the rabbits are immortal and incest is OK!
Fib (n2) Fib (n1) Fib (n)
6 Why is this so interesting
Appears in so many forms of nature for example the number of petals on flowers
(2) Crown of Thorns (3) Iris (5) Pinks (1) White Calla (13) Cineraria (21) Aster (34) Daisy (8) Coreopsis 7 We can see spirals are consecutive Fibonacci numbers The same can be said about the flower only it has more spirals and is harder to show! 8 Is that really amazing NO! But this is 9 Mathematically find the limit
We can find the nth term of the sequence by Binets formula
Fib (n) (1 / v5)((1 v5)/2)n ((1 - v5)/2)n
Using this we can find the limit of the sequence as
Now we need to find the limit of the sequence as n8 of
Fib (n 1) a
a (1 / v5)((1 v5)/2)(n1) ((1 - v5)/2)(n1)
(1 / v5)((1 v5)/2)n ((1 - v5)/2)n
Divide everything through by (1 / v5)
a ((1 v5)/2)(n1) ((1 - v5)/2)(n1)
((1 v5)/2)n ((1 - v5)/2)n
Divide through by (1 v5)/2)n
a ((1 v5)/2)(n1)/((1 v5)/2)n((1 - v5)/2)(n1)/((1 v5)/2)n
1 ((1 - v5)/2)n/ ((1 v5)/2)n
Fib (n 1) (1 v5) as n 8
Fib (n) 2
Which approximately 1.618033989
11 What relevance does this have
Golden Rectangle (ratio of sides 1Phi)
12 So where does this apply to our examples at the start
They all exhibit the Golden Ratio!
13 Spira Mirabilis Even the human ear exhibits this spiral 14 Class Volunteer
Arm length 78.5cm
Forearm length 50cm
Arm length 1.57
distance to belly button 111cm
distance to belly button
15 Who would be you valentine Colin Farrell Matthew McConaughey Tom Cruise Harrison Ford 1.45 1.55 1.35 1.60 16 Conclusion
Phenomenal interest to human beings
There has been no actual formula constructed to show that
beauty k x Golden ratio
Although there does appear to be a relationship
There is no formal accepted description of aesthetic judgement in mathematics (Livio 2002).
17 Any questions
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