Unravelling the Fibonacci Sequence

-Hetvi Sanghavi Vora

Fibonacci believed to be some secret code that manifests the architecture of the universe, is a sequence of numbers in which a given number is the result of adding the two numbers before it. Starting with 0 and 1, the series proliferates into 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144 and so on. What’s so special about these series? This is the pattern that functions the creation of living beings in this universe. Take a look around at Nature. You will see this pattern appearing in flowers as it is easy to notice.

Fibonacci Spiral

When squares with sides as the units in the Fibonacci Sequence like squares of units 1, 1, 2, 3, 5, and 8 are constructed and placed beside each other in a manner that makes a rectangle would result in a spiral when diagonal ends are connected with the help of a compass forming a continuous length in the form of a spiral which is called as the golden spiral.

When you divide two adjacent numbers in the Fibonacci sequence, you get either 1.61 or 0.1618. For example, 89/55 is 1.618. In the Fibonacci sequence, any given number is approximately 1.618 times the preceding number, ignoring the first few numbers. Each number is also 0.618 of the number to the right of it, again ignoring the first few numbers in the sequence. These are called Phi and phi respectively. Also called the Golden Ratio or the Divine Proportion, in mathematics, the irrational number (1+√5)/2, is often denoted by the Greek letter ϕ or τ, which is approximately equal to 1.618.  We find the golden ratio when we divide a line into two parts so that the long part divided by the short part is also equal to the whole length divided by the long part.

(a+b)/a = a/b = Phi = ϕ is the formula for golden ratio.

 Fibonacci in Flowers-

The pattern of seeds of a sunflower forms the classic Fibonacci Spiral. The number of petals of flowers is always a Fibonacci number. If you are wondering how Fibonacci flowers create such perfect floret arrangements, then the answer lies in the plant hormone called ‘auxin’. The growth hormone, auxin, helps in the growth and development of leaves, flowers, stems, and other plant parts. The plant grew where the auxin flows and interacts with other proteins. And since the hormone flows in the plant in a spiral direction, the plant grows spirally, leading to “Fibonacci spirals” in sunflowers.

Lilies and Irises have 3 petals, buttercups and wild roses have 5 petals, delphiniums 8 whereas Michaelmas daisies have 55 or 89 petals. All these are Fibonacci numbers.

Mathematician Leonardo Bonacci discovered the Fibonacci sequence. The mathematical equation it is Xn+2= Xn+1 + Xn. We see this pattern appearing in snail shells, pineapples, pinecones as well as images of nebulas and galaxies that depict the Fibonacci pattern. The cochlea of the inner ear forms a Golden Spiral according to this pattern.

Fibonacci in Music

The Fibonacci sequence of numbers and the golden ratio are manifested in music widely. The numbers are present in the octave, the foundational unit of melody and harmony. Stradivarius used the golden ratio to make the greatest string instruments ever created. Therefore it is the universal matrix on the fabric of which nature is created.

The Fibonacci sequence nicknamed ‘nature’s code’, was used by Mozart as he arranged his piano sonatas so that the number of bars in the development and recapitulation divided by the number of bars in the exposition would equal approximately 1.618, the Golden Ratio. In one sonata by Mozart, The exposition consists of 38 bars and the development and recapitulation consist of 62. The first movement as a whole consists of 100 bars. Here as well, 62 divided by 38 equals 1.63 (approximately the Golden Ratio).

Out of the13 keys in an octave, a major scale can be created in that octave using 8 notes. Within that major scale the 1st, 3rd, and 5th notes create the most basic major chord. Starting with Chord C Major that is made up of Sa Ga Pa or Do Mi So till Chord B Major the same numbers of 1, 3, 5, 8 are followed. Also, the ratio of white to black keys within an octave is 8:5. The ratio of which is 1.6.  The pentatonic scale has five notes, the Diatonic scale has eight notes, and the Chromatic scale has thirteen notes which are all Fibonacci numbers.

Another relation to music from the Fibonacci sequence is how musical instruments are created. Φ was used to create Stradivarius violins, the most sought-after and expensive violins in the world, one of which sold for $3.6 million (Stradivarius Violin Price, 2020). The violinmaker ensured the proportion of the neck, pegbox, and scroll to the body of the violin (upper bout, waist, and lower bout) achieves the ratio. Also, subdivisions of the instrument – waist to the upper bout, waist, and upper bout to those sections plus the neck – meet the 1.6 ratios as well.

Fibonacci in Art and Architecture

Well, the sequences have been found widely in architecture as well as medieval sculptures. The human face, by averages, follows the Golden Ratio in where the eyes, nose, and chin are positioned. The proportion of DNA molecules, 34 by 21 angstroms, fit the ratio as well. The pyramids of Egypt follow some unknown mysteries as per the Golden Ratio. This is also how many companies design their logos. Apple’s logo is a classic example of the same. This is used in art, architecture, and sculptures for ages.

The pentagram which is famous as a magical or holy symbol has the Golden Ratio in it.

Conclusion

Thus the Fibonacci sequence has been observed in many aspects such as nature, art, architecture, music as well as nebulas and galaxies. This is Mysterious Maths which is cherished by nature. That’s not all the ways this proportion has been used in trading and coding has been exceptional too. Have you encountered this divine proportion?

References-

  1. https://www.livescience.com/37470-fibonacci-sequence.html
  2. https://www.classicfm.com/discover-music/fibonacci-sequence-in-music/#:~:text=The%20Fibonacci%20Sequence%20in%20music&text=%E2%80%93%20An%20octave%20on%20the%20piano,and%20five%20are%20black%20keys.&text=%E2%80%93%20In%20a%20scale%2C%20the%20dominant,that%20make%20up%20the%20octave.&text=These%20are%20all%20numbers%20in,%2C%205%2C%208%2C%2013
  3. https://www.classicfm.com/discover-music/fibonacci-sequence-in-music/
  4. https://etd.ohiolink.edu/apexprod/rws_etd/send_file/send?accession=oduhonors1620086748612102&disposition=inline
  5. https://auralcrave.com/en/2020/06/28/the-golden-ratio-in-music-the-songs-of-fibonacci-sequence/
  6. https://www.mathsisfun.com/numbers/fibonacci-sequence.html
  7. https://mathworld.wolfram.com/FibonacciNumber.html
  8. https://www.classicfm.com/discover-music/fibonacci-sequence-in-music/
  9. https://www.benningviolins.com/fibonacci-series-and-stradivarius-instruments.html#:~:text=It%20also%20was%20the%20basis,of%20the%20two%20preceding%20it
  10. http://www.eniscuola.net/en/2016/06/27/the-numbers-of-nature-the-fibonacci-sequence/
  11. http://www.eniscuola.net/wp-content/uploads/2016/06/Pdf_TheNumbersOfNature.pdf
  12. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
  13. https://www.pansymaiden.com/flowers/types/fibonacci-flowers/
  14. https://www.rnz.co.nz/national/programmes/afternoons/audio/2018690395/maths-and-dance-combine-in-the-fibonacci
  15. https://www.creativebloq.com/design/designers-guide-golden-ratio-12121546
  16. https://www.nationalgeographic.org/media/golden-ratio/ 

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