The ratio of successive pairs tends to the so-called golden section (GS or PHI ) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.
It is quite amazing that the Fibonacci number patterns occur so frequently in nature ( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature".
Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:
- 3 petals: lily, iris
- 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
- 8 petals: delphiniums
- 13 petals: ragwort, corn marigold, cineraria,
- 21 petals: aster, black-eyed susan, chicory
- 34 petals: plantain, pyrethrum
- 55, 89 petals: michaelmas daisies, the asteraceae family
Plants do not know about this sequence - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.
Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.
So nature isn't trying to use the Fibonacci numbers: they are appearing 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.
The basic idea is that the position of each new growth is about 222.5 degrees away 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 . . . .
( For further reading: http://www.world-mysteries.com/sci_17.htm )
The number PHI derived from the Fibonacci sequence – a progression famous not only because the sum of adjacent terms equaled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1.618 – PHI. Despite PHI’s seemingly mystical mathematical origins, the mind –boggling aspect of PHI is its role as a fundamental building block in nature. The ancients assumed the number PHI must have been preordained by the Creator of the universe. Early scientists heralded one-point-six-one-eight as the Divine Proportion”
The Devine Proportion has been extensively used by artists and musicians world wide. Leonardo Da Vinci’s famous male nude – The Virtruvian Man – is a brilliant example.
The artwork by Michelangelo, Da Vinci, and many others, adhered to the Divine Proportion meticulously. PHI is used in the architectural dimensions of the Greek Parthenon, and the pyramids of
Nature’ Laws embraces the most important of all elements, timing. Nature’s Laws is not a system, or method of playing the market, but it is a phenomenon, which appears to mark the progress of all human activities. Its application to forecasting is revolutionary.
All human activities have three distinctive features, pattern, time and ratio, all of which observe the Fibonacci summation series.”
Later I found that the basis of my discoveries was a Law of Nature known to the designers of the Great Pyramid “Gizeh”, which may have been constructed 5000 years ago.
The dancing of Shiva is the dancing of universe; the ceaseless flow of energy going through an infinite variety of patterns that melt in to one another. Indian artists of the tenth and twelfth centuries have represented Shiva’s cosmic dance in magnificent bronze sculptures of dancing figures with four arms whose superbly balanced and yet dynamic gestures express the rhythm and unity of Life. The upper right hand of Shiva holds a drum to symbolize the primal sound of creation, the upper left bears a tongue of flame, the element of destruction. The balance of the two hands represents the dynamic balance of creation and destruction in the world, accentuated further by the Dancer’s calm and detached face in the centre of the two hands, in which the polarity of creation and destruction is dissolved and transcended. The second right hand is raised in the sign of ‘do not fear’, symbolizing maintainance, protection and peace.
As of Nandi you can also see the basic pattern of Elliot’s Waves in the dancing Nataraja idol shown above. First two waves in the left leg, the powerful third wave through the body from the right side of the hip to the left shoulder (path of Poonool), & Fourth and Fifth waves in the left hand. After the completion of the Fifth wave there is flame symbolizing the destruction thereafter.