the sunflower spiral and the fibonacci...
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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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![Page 2: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/2.jpg)
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'
![Page 3: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/3.jpg)
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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![Page 4: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/4.jpg)
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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![Page 5: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/5.jpg)
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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![Page 6: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/6.jpg)
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/
![Page 19: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/19.jpg)
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.
![Page 24: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/24.jpg)
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.
![Page 25: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/25.jpg)
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.
![Page 26: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/26.jpg)
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.
![Page 27: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/27.jpg)
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.
![Page 28: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/28.jpg)
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.
![Page 29: The Sunflower Spiral and the Fibonacci Metricmath.okstate.edu/people/segerman/talks/sunflower_spiral_talk.pdf · 1 2 3 4 5 6 7 9 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa38eaafbaf7ae63c9da7/html5/thumbnails/29.jpg)
Thanks!