Maths 190 Lecture 3
I Topic for today: Numerical Patterns inNature
I Vitally important question:
What do rabbits, pineapples, daisys,sunflowers and pinecones have incommon?
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Pineapples and pinecones
I Look carefully at your pineapple.
I How many spirals does your pineapple have?
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Fibonacci Numbers
I 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
I Call these F (1), F (2), F (3), etc
I ThenF (n) + F (n + 1) = F (n + 2)
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She loves me, she loves me not...?
I Pulling petals off a flower, we’d like to know our chances ofgetting an odd number - “She loves me”!
I Most flowers have a Fibonacci number of petals.
I How many of the Fibonacci numbers are odd?
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Which are odd?
I 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, . . .
I Looks like every third one is even.
I How can we show this is always true?
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What about rabbits?
I Suppose we start with one breeding pair of rabbits.
I Each pair of rabbits produces one pair of offspring eachmonth.
I It takes the rabbits one month to mature.
I How many rabbits do we have?
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Ratio of F (n)
I What is the ratio F (n + 1)/F (n) asn gets large?
I Seems to converge...
I The Golden Ratio ψ.
11 = 1
21 = 2
32 = 1.5
53 = 1.6667
85 = 1.6
138 =
2113 =
3421 =
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Fibonacci Building Blocks
I Claim: Every natural number is either a Fibonacci number, orcan be written as the sum of Fibonacci numbers.
I Try it!
I In fact, every natural number can be written as the sum ofnon-consecutive Fibonacci numbers.
I Use a “greedy” algorithm:I subtract the biggest possible Fibonacci number to get a
smaller number.I repeat as required.
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Important ideas from today:
I Certain patterns, and sequences of number reappearthroughout nature. One common one is the Fibonaccisequence.
I Rabbits breeding, the numbers of spirals on pinecones anddaisies, all follow the Fibonacci sequence.
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