Golden Mean RatioFirst they ignore you. Then they laugh at you. Then they fight you. Then you win. - Mohandas Gandhi.
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Fibonacci Sequence

Phi Golden Ratio or Golden Mean Ratio

The Phi Golden Ratio or Golden Mean Ratio was first published by Fibonacci, apparently as a simple exercise without realizing its far reaching implications in nature. Fibonacci's example was:

A pair of rabbits are put in a field and if rabbits take a month to become mature, then produce a new pair every month after that, how many pairs will there be in twelve months time?

The answer to this riddle is the famous Fibonacci Sequence, which represents the number of rabbits each month. It looks like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. Starting with 1, 1, the next value is simply the addition of the two previous values. This series of values wasn't known as Fibonancci Series until a French matematician, Edouard Lucas (1842-1891), found many more applications for the series.


Fibonacci Sequence in Nature

Fibonacci's example problem has a wide variety of applications in nature since it illustrates the growth pattern of many species including trees, seeds, honeybee genetics, etc.


Optimal Packing with the Fibonacci Series

One of the places that the Fibonacci Series shows up routinely in nature is in the growth of seeds on pine cones and some flowers. The number of seeds in each row is increased by the Fibonacci series. This series allows for optimal packing of seeds in a spiral manner.

Pine Cone
Fibonacci Sequence

New seeds on a sunflower are only formed at the very center. Yet, when you see a fully grown sunflower, you'll notice that it has hundreds of seeds that are all evenly spaced throughout the face. Look closely and you can see the spiral of seeds. You'll notice the same in the daisy. The magic of seed and plants growing like this comes from the magic in the Fibonacci series and Phi golden mean.

Phi Golden Ratio Fibonacci Spiral



Many flowers and plants will have some Fibonacci number of pedals or leaves.

For example:


# Of Petals Flower Varieties
3 Lily, Iris (Sometimes 6 formed in two pairs of 3)
5 Buttercup, Wild Rose, Larkspur, Columbine (aquilegia)
8 Delphiniums
13 Corn, Marigolds, Cineraria, and Some Daisies
21 Aster, Black-eyed Susan, and Chicory
34 Plantain, Pyrethrum
55, 89 Michaelmas Daisies, the Asteraceae family


Examples of the Fibonacci Sequence in the Real World

Leonardo Fibonacci observed what was later called the Fibonacci Sequence in the early 13th century. Golden Mean Ratio
Leonardo Fibonacci observed what was later called the Fibonacci Sequence in the early 13th century. Fibonacci Series
Fibonacci Sequence Fibonacci Spiral
Leonardo Fibonacci observed what was later called the Fibonacci Sequence in the early 13th century. Fibonacci Spiral
Fibonacci Sequence Fibonacci Series
Phi Golden Ratio  



Longitudinal Wavelength Sound Waves Pitch and Frequency Speed of Sound Doppler Effect Sound Intensity and Decibels Sound Wave Interference Beat Frequencies Binaural Beat Frequencies Sound Resonance and Natural Resonant Frequency Natural Resonance Quality (Q) Forced Vibration Frequency Entrainment Vibrational Modes Standing Waves Law of Octaves Psychoacoustics Tacoma Narrows Bridge Schumann Resonance Animal BioAcoustics More on Sound


Law Of Octaves Sound Harmonics Western Musical Chords Musical Scales Musical Intervals Musical Mathematical Terminology Music of the Spheres Fibonacci Sequence Circle of Fifths Pythagorean Comma


Drum Vibrational Modes


Aristotle Copernicus Einstein Fibonacci Hermann von Helmholtz Kepler Sir Isaac Newton Max Planck Ptolemy Pythagoras Thomas Young
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