Patterns in Nature â€œNature geometrizes universally in all her manifestations.â€? H.P. Blavatsky

Anhalt University - D.I.A.- Professor: Christos Passas - Student: Deborah Kaiser

Phi and Fibonacci

Phi and Fibonacci

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Fibonacci sequence It begins with 0 and 1, and each subsequent number is generated by adding the two preceding numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 .

1/1 2/1 3/2 5/3

8/5

13/8 21/13

34/21

55/34

89/55

144/8 9

1.0 2.0 1.5 1.66 1.60 1.62 1.61538 1.61904 1.61764 1.61818 1.6179 6 0 5 5 8 7 2 78

A

B

C

AB/BC = AC/AB = ö = (√5+1)/2 ö² = ö+1 1/ö = ö-1

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The large rectangle is a golden rectangle, meaning that its sides are in the proportion 1.000:1.618. Connecting the successive points where the 'whirling squares' divide the sides of the rectangles in golden ratios produces a logarithmic spiral, which is found in many natural forms. A similar spiral can be generated from a golden triangle (an isosceles triangle whose sides are in the golden ratio), by repeatedly bisecting one of the angles to generate a smaller golden triangle.

Underlying Order in Nature

Underlying Order in Nature Many of nature's patterns are related to the Fibonacci numbers and golden ratio. The golden spiral, a logarithmic or equiangular spiralis,is found in unicellular foraminifera, sunflowers, seashells, animal horns and tusks, beaks and claws, whirlpools, hurricanes, and spiral galaxies. These arrangements maximize the space for each leaf and the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. Fibonacci numbers appear then as a by-product of a deeper physical process. That is why the spirals are imperfect. The plant is responding to physical constraints, not to a mathematical rule.

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The golden angle The plant uses the golden angle because it packs the most into the smallest area.

Golden angle

Plus 0.1 %

Minus 0.1 %

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The position of each new growth is about 222.5 degrees away 08 / 16 from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . . If we call the golden section GS, then we have 1 / GS = GS / (1 - GS) = 1.618033989 . . . . If we call the golden angle GA, then we have 360 / GA = GA / (360 -GA) = 1 / GS.

A ram's heavy corkscrewed horns keep a stable centre of 09 / 16 gravity as they grow. An equiangular spiral does not alter its shape as its size increases

10 / 16 Fibonacci helices can be seen on pine-cones, pineapples, and teazles. They can be seen in the phyllotaxis of many plants.

Fibonacci fraction: leaves/spirals

5/2

5/3

8/3

8/5

To get from the topmost leaf to the last of the 5 leaves of the plant on the left takes 2 anticlockwise turns or 3 clockwise turns. 2, 3 and 5 are three consecutive Fibonacci numbers. For the plant on the bottom, it takes 3 anticlockwise rotations or 5 clockwise rotations to pass 8 leaves. Again, 3, 5 and 8 are consecutive Fibonacci numbers.

Angle between consecutive leaves

144째

216째

135째

225째

Leaf arrangement

Apple, cherry, apricot, oak, cypress, poplar

Holly, pear, spruce, various beans

An estimated 90% of all plants display this pattern.

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The program chooses random numbers, from which it generates sets of 500 points, each set having a particular angular spacing, chosen at random between 0 and 360 degrees. Then it uses the golden angle so that you can see how no obvious pattern emerges, and how the space is filled without apparent bias. That is very odd, in that a random filling is in a sense more symmetrical than the rational ones, in that it looks the same everywhere. On the other hand, a random set has no symmetry, in that it is unique everywhere. No operation can superimpose it on itself except the identity operator.

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Patterns

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