# Free The Golden Ratio/Fibonacci Sequence In Nature Essay Example

The golden ratio is a number, represented by the symbol φ, such that between φ and 1, along with 1 and 1- φ, the ratio is the same. It is an irrational number explicitly created by the formula 1+√52 = 1.618033988 (Beck and Geoghegan). Like fractals, the golden ratio unifies several different areas of mathematics together. Interestingly enough, it is also found in many places in nature. The golden ratio has been detected even at the quantum level, where liked magnetic atoms appear to vibrate at frequencies described by φ.

The Fibonacci sequence is a number sequence, essentially 1,1,2,3,5,8,13,21,34,55,89,144 This sequence of digits is found by adding the two numbers before, resulting in a very unique sequence of numbers that shows up periodically, in music, art, and nature as well, such as the petals of flowers or the shapes of pinecones. They are named after a the very famous Italian mathematician Leonardo of Pisa, notoriously known as Fibonacci, the first person to actually publish a book in Western Europe that used the Indian numerals 9, 8, 7, 6, 5, 4, 3, 2, and 1 (Mitchison).

The Fibonacci sequence and the golden ratio, two seemingly unrelated topics, in fact, produce the exact same number, this is a very, very rare occurrence, taking into account that this number (or ratio) is an irrational integer.

There exists a big amount of divergent scientific opinion regarding the legitimacy of golden ratio and when compared to other numbers, the occurrence of golden ratio. This leads to the question of whether the evidence of occurrence of golden ratio in so many aspects of the universe is just a confirmation bias or not. The world as observed by human beings is seemingly full of numbers and of patterns which human beings either learn about by direct observation or develop a model about which is later confirmed by observation. In this way, numbers play an important role in shaping the world around human beings. However humans are constantly spending collective and individual efforts to find an explanation which is coherent and logical and can be applied to anything which is not yet known, in order to deduce or infer knowledge about it. In this manner human beings develop an interest and subjective pleasure about objective reality of nature whereby they assign aesthetic value or otherwise non-numerical prestige in phenomenon which they perceive as such. Human beings tend to notice the numbers and patterns when they are looking for it in open nature, but more often than not when they are not looking for it actively, they tend to ignore the occurrences especially if an apparent pattern is not obvious to them right away.

However, the prevalence of the Fibonacci number has more to do with maximum biological efficiency and natural growth (Douady and Couder). The function of growing efficiently is in fact fairly straight forward, all explained by divergence angles and phyllotactic ratios. Phyllotaxis is a word of Greek language origin which means, to an approximation, ‘leaf arrangement’. This term was first used by and possibly created by a Swiss Naturalist of eighteenth century, Charles Bonnet. A handful of decades later, in 1830, two scientists observed in nature the phenomenon of each new leaf and recorded geometrical data about the growth. They observed that each new leaf grows at an angle of about 137.5 degrees between the consecutive leaves.

The widely accepted botanical definition of phyllotactic ratio is that it’s the amount of divergence from previous leaf for which each newly sprouting leaf separates. If the divergence angle is approximately 137.5 degrees or in ratios around 0.618, this value is to be subtracted from 1 to get 1ϕ2, or in simpler terms, the ratio of two very large consecutive Fibonacci numbers:

This arrangement of leaves on a stem, the phyllotaxis, and the arrangement of some flowers’ seeds, often display periodical spiraling patterns. When the number of spirals in either of the directions are counted, clockwise or counterclockwise, a familiar set of numbers like 13 and 21 is often encountered, along with 5,8,34,55, and 89.

As seen in the image above, the packing of seeds on the head of a sunflower features a series of clockwise and counterclockwise spirals (Vogel). These spirals occur in successive Fibonacci numbers.

Fibonacci sequence shows remarkable properties in the fields of natural sciences and pure mathematics in unexpected ways. One such occurrence of Fibonacci sequence in mathematics is exhibited in the divisibility properties of co-primes (Glaister). Any three Fibonacci numbers which are consecutive are co-primes, which means that for every n where Fn represents a Fibonacci number, the following relationship is true (Mathforum.org):

gcdFn, Fn+1=gcdFn+1, Fn+2=gcdFn, Fn+2=1

Thus every prime number, ρ, can be shown to divide a Fibonacci number where the number can be determined by the value of that prime number modulo 5. In order to show this, some background information regarding modular arithmetic is needed. If ρ is equal to 1 mod 5 or 4 mod 5, then ρ divides F ρ-1. If ρ is equal to 2 mod 5 or 3 mod 5, then ρ divides F ρ+1. And the remaining condition is that if ρ is 5, then it divides Fρ. These results can be compiled in the following table:

## So even in the domain of pure mathematics, the occurrence of golden ration and Fibonacci sequence is prominent.

Another phenomenon in nature where golden ratio is observed is in the growth of many organisms. The most well-known example of such an organism is mollusk Nautilus, the shell of which is often shown in relation with golden ratio. Given below is a picture of a cross section of Nautilus shell. The points marked with blue dots correspond to growth of new sections which follows the golden ratio closely.

Another common phenomenon in nature is lightening where at first glance there is no sign of golden ratio. However if lightning strikes a rock or a boulder, and the cross section of that rock is studied, a fractal-like pattern can be found which follows the golden ratio very closely. To see that, below is given a cross section from an acrylic polymer block.

## This polymer block was subjected to extremely high voltage which resulted in the emerging of this pattern shown.

Snowflakes are all said to be unique in their appearance. However the symmetry of each snowflake is independent of its uniqueness and gives amazing insight to the working of nature and how patters which fold over or repeat themselves tend to closely mimic the golden ratio and Fibonacci sequence. The symmetry of all snowflakes stems from the fact that ice has six-fold symmetry in its structure. This six-fold symmetry is inherent to ice and so every snowflake in the world produced through natural processes exhibits the same symmetry. Fractal nature of snowflakes makes this symmetry repeat itself over and over again. And the complexity of mathematics involved in this kind of symmetry calculation makes it extremely difficult to mimic real snowflakes in a video simulation. And as the number of snowflakes in the simulation increase, amount of computer power required to produce that simulation increases explosively.

As stated in the introduction, golden ratio is found universally, from cosmological sizes of galaxies and star systems to quantum mechanics and resonations of similar atoms. The golden ratio was first discovered in quantum world in 2010 when researchers from the Helmholtz-Zentrum Berlin observed a nanoscale symmetry in solid state matter. This observation came as a result of combined efforts of these researchers collaborating with people from Oxford University and Rutherford Appleton Laboratory. To understand this experiment, some discussion on how atoms work on an extremely small scale is required. When atoms are studied at such a minute scale, they come under the influence of Uncertainty Principle where the direct observations of these atoms are physically impossible to be arbitrarily accurate. So to study these effects, researchers have successfully made very thin chains of atoms which are just one atom thick. These chains are made from magnetically susceptible atoms and when the researchers apply a magnetic field to such structures, the atoms realign to produce a new state called quantum critical. In this new critical state, atoms exhibit a fractal pattern which gives frequencies distributed in the ratio very close to 1.618. The difference between golden ratio and such frequencies of quantum critical can be explained by background radiation and other side effects.

## References

Helmholtz Association of German Research Centres,. 'Golden Ratio Discovered In Quantum World: Hidden Symmetry Observed For The First Time In Solid State Matter'. ScienceDaily. N.p., 2010. Web. 10 Mar. 2015.

Knott, R. 'The Fibonacci Numbers And Golden Section In Nature - 1'. Maths.surrey.ac.uk. N.p., 2015. Web. 10 Mar. 2015.

Nelson, Jon. 'The Six-Fold Nature Of Snow'. Storyofsnow.com. N.p., 2015. Web. 10 Mar. 2015.

Williams, H. C. 'A Note On The Fibonacci Quotient ". Canadian Mathematical Bulletin 25.3 (1982): 366-370. Web.

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