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The Sunflower Spiral and the Fibonacci Metric
Henry Segermanhttp://www.segerman.org
Department of Mathematics and StatisticsThe University of Melbourne
July 28th 2010
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The Sunflower SpiralLet S(n) = (r(n), θ(n)) = (
√n, 2πϕn), where
ϕ =√
5−12 = Φ− 1 ≈ 0.618, and Φ =
√5+12 is the golden ratio.
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2¼'
The Sunflower SpiralLet S(n) = (r(n), θ(n)) = (
√n, 2πϕn), where
ϕ =√
5−12 = Φ− 1 ≈ 0.618, and Φ =
√5+12 is the golden ratio.
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2
The Sunflower SpiralLet S(n) = (r(n), θ(n)) = (
√n, 2πϕn), where
ϕ =√
5−12 = Φ− 1 ≈ 0.618, and Φ =
√5+12 is the golden ratio.
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2
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The Sunflower SpiralLet S(n) = (r(n), θ(n)) = (
√n, 2πϕn), where
ϕ =√
5−12 = Φ− 1 ≈ 0.618, and Φ =
√5+12 is the golden ratio.
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The Sunflower SpiralLet S(n) = (r(n), θ(n)) = (
√n, 2πϕn), where
ϕ =√
5−12 = Φ− 1 ≈ 0.618, and Φ =
√5+12 is the golden ratio.
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This sequence of points models many patterns in nature, inparticular the florets on a sunflower head.
Photo credit: http://www.flickr.com/photos/lucapost/694780262/
Fibonacci Metric
Let M : N→ N be a function, M(n) is the minimal number ofFibonacci numbers Fi needed in order to sum to n.
1 = 1 M(1) = 12 = 2 M(2) = 13 = 3 M(3) = 14 = 3 + 1 M(4) = 25 = 5 M(5) = 16 = 5 + 1 M(6) = 27 = 5 + 2 M(7) = 28 = 8 M(8) = 19 = 8 + 1 M(9) = 2
10 = 8 + 2 M(10) = 211 = 8 + 3 M(11) = 212 = 8 + 3 + 1 M(12) = 3
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Where do the patterns come from?
1. Radial spokes2. Circular tree rings
For the spoke at θ = 0:
Fk = Φk−(1−Φ)k√5
⇓Fk − ϕFk+1 = (−ϕ)k+1
⇓θ(Fk+1) = 2πϕFk+1
= 2π(Fk − (−ϕ)k+1)
(−ϕ)k+1 is small, and so for large k , θ(Fk+1) is almost a multipleof 2π.
Thus the Fibonacci numbers themselves are near θ = 0.
Where do the patterns come from?
1. Radial spokes2. Circular tree rings
For the spoke at θ = 0:
Fk = Φk−(1−Φ)k√5
⇓Fk − ϕFk+1 = (−ϕ)k+1
⇓θ(Fk+1) = 2πϕFk+1
= 2π(Fk − (−ϕ)k+1)
(−ϕ)k+1 is small, and so for large k , θ(Fk+1) is almost a multipleof 2π.
Thus the Fibonacci numbers themselves are near θ = 0.
Where do the patterns come from?
1. Radial spokes2. Circular tree rings
For the spoke at θ = 0:
Fk = Φk−(1−Φ)k√5
⇓Fk − ϕFk+1 = (−ϕ)k+1
⇓θ(Fk+1) = 2πϕFk+1
= 2π(Fk − (−ϕ)k+1)
(−ϕ)k+1 is small, and so for large k , θ(Fk+1) is almost a multipleof 2π.
Thus the Fibonacci numbers themselves are near θ = 0.
Where do the patterns come from?
Sums of Fibonacci numbers haveangles the sum of the angles of theFibonacci numbers, so sums of asmall number of large Fibonaccinumbers are also near θ = 0. Thismakes up other points of the θ = 0spoke.
Other spokes are rotations of theθ = 0 spoke, formed by adding asmall number to all of the numbersin the θ = 0 spoke.
For the tree rings: Just after a large number m with small M(m),there will be many numbers n for which the minimal M(n) isachieved using m and some small number of additional Fibonaccinumbers, because n −m is small.
Where do the patterns come from?
Sums of Fibonacci numbers haveangles the sum of the angles of theFibonacci numbers, so sums of asmall number of large Fibonaccinumbers are also near θ = 0. Thismakes up other points of the θ = 0spoke.
Other spokes are rotations of theθ = 0 spoke, formed by adding asmall number to all of the numbersin the θ = 0 spoke.
For the tree rings: Just after a large number m with small M(m),there will be many numbers n for which the minimal M(n) isachieved using m and some small number of additional Fibonaccinumbers, because n −m is small.
Where do the patterns come from?
Sums of Fibonacci numbers haveangles the sum of the angles of theFibonacci numbers, so sums of asmall number of large Fibonaccinumbers are also near θ = 0. Thismakes up other points of the θ = 0spoke.
Other spokes are rotations of theθ = 0 spoke, formed by adding asmall number to all of the numbersin the θ = 0 spoke.
For the tree rings: Just after a large number m with small M(m),there will be many numbers n for which the minimal M(n) isachieved using m and some small number of additional Fibonaccinumbers, because n −m is small.
Thanks!
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