![]() This problem is more famously termed as “the problem of division of a line segment in extreme and mean ratio” (Benavoli et al., 2019). The Golden Ratio is believed to have come from Euclid, particularly from his famous book Elements, where he tried to solve a problem concerning the division of a line segment into two unequal parts and noted the size ratio of the resultant parts. The term ‘Golden Ratio’ is often used interchangeably with the term ‘Golden Section,’ although some scholars have designated the term ‘Golden Section’ to be the reciprocal of the Golden Ratio. The Golden Ratio is a concept that bears a strong connection to the Fibonacci Sequence. The Fibonacci sequence can be written as. Where Fn represents the nth Fibonacci number. The Fibonacci sequence, and the associated Fibonacci numbers, are defined by the following equation:į n = F n-1 + F n-2 for all n ≥ 3 where F 1 = 1 F 2 = 1 One source of natural patterns is the Fibonacci sequence and the associated Fibonacci numbers. The Fibonacci sequence has proven to be ubiquitous, not only in the natural world but also in some structures within the human body like the cochlea (Pietsch et al., 2017). Throughout history, human beings have engrossed themselves with the unabating search for patterns in the physical world. The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. ![]() Where We Can Find Equiangular Spirals (or Logarithmic Spirals) The Fibonacci Numbers and the Golden Section in Nature This will be explored in a future article It is another way that Phi and the related Fibonacci numbers are seen in nature. So this is a way also to arrange leaves on a twig or twigs on a branch so that the leaves do not shade each other too much. The reason the packing is optimal is that the amount of overlap is minimized. This same idea works for arranging leaves in a way that they shade each other the least. It is an optimal way to place seeds on a flower head or scales on a pine cone. So this is a way to approximate the theoretical best packing: form 3 seeds every 5 turns, or 5 seeds every 8 turns, or 8 every 13 turns, and so on. You will find that the quotient comes closer and closer to Phi as you use bigger F-numbers. Make fractions that have a Fibonacci number in the numerator and the previous Fibonacci number in the denominator: 5/3, 8/5, 13/8, 21/13, and so on. The Fibonacci Numbers Approximate the Golden Mean Notice that in the last example the “seeds” could be pushed closer and fill the space more efficiently. Notice a decrease in overlap and increased use of space in the last example.Ī diagram showing the progression of 1 to 5 “seeds” per rotation to phi. In the illustration (click to enlarge) the examples progress from 1/2 to 1/6 turn per cell, and then Phi. This will give the least amount of overlap and the best packing. So the ideal pattern would be to produce a new seed every 137.5 degrees of rotation. If you divide a circle by phi, you get 2 angles, the smaller of which is about 137.5 degrees. One such irrational number is Phi, the golden ratio, 1.61803…. The theoretical best fraction would be an irrational number which cannot be expressed exactly by whole numbers. Any rational fraction (the top and bottom are whole numbers) will give this effect. Most of the space between each line of cells is wasted. If cells grow every ½ turn then they pile up in 2 lines, 1/3 turn would give 3 lines, and so on. How much of a circle should the growth center rotate before growing a new cell? If it rotates a whole turn then the cells still pile up on each other in 1 line. ![]() As they form, the center of growth is rotating so that the cells don’t end up on top of each other. Plants grow from active tissue called the meristem, most often at the tip of the branches. This optimal packing means that the plant can build a smaller structure to hold them. The pine scales and seeds are arranged in a way that requires minimum space. The pattern represents an optimization of resources. This gives 1, 1, 2, 3, 5, and so on as above. This series is formed from the starting numbers 1, 1, and then adding together the last 2 numbers to get the next one. The sequence 5, 8, 13, 21, 34, and 55 are members of the Fibonacci series. The numbers occur in these pairs more often than not. How many in each direction? There can be 5 and 8, 8 and 13, 21 and 34, 34 and 55, and sometimes more. Fibonacci Numbers appear when you count the spirals The spirals can be seen in both clockwise and counterclockwise directions. The scales of the cones and the seeds in the flower trace graceful spirals radiating out from the center. ![]() Pine cones and flower heads of the composite family of flowers both show a similar pattern. Fibonacci numbers can be found in many remarkable patterns in nature.
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