The Fibonacci sequence is possibly the most simple recurrence
relation occurring in nature. It is 0,1,1,2,3,5,8,13,21,34,55,89, 144…
each number equals the sum of the two numbers before it, and the
difference of the two numbers succeeding it. It is an infinite sequence
which goes on forever as it develops.
The Golden Ratio/Divine Ratio or Golden Mean
The quotient of any Fibonacci number and it’s predecessor approaches Phi, represented as ϕ (1.618), the Golden ratio. The Golden Ratio is best understood geometrically by the golden rectangle. A rectangle unevenly divided resulting into one square and one rectangle, the square’s sides would have the ratio of 1:1, and the new rectangle would be exactly proportionate to the original rectangle – 1:1.618.
This iteration can continue both ways, infinitely. If you plot a quarter circle inside each of the squares as they reiterate, the golden spiral is formed. The golden spiral is possibly the most simple mathematic pattern that occurs in nature like shells of snails, sea shells, horns, flowers, plants. Numbers are only what we use to organize quantitative information.
The Golden Ratio can be applied to any number of geometric forms including circles, triangles, pyramids, prisms, and polygons. The golden ratio is formed by thirds within thirds, sixths, the connection between two and three, including every even and odd number itself. The ratio itself represents the transcendence of numbers, understanding our world is not numbers, but what numbers represent. Through the spiral, the ratio illustrates how the numbers, all quantities, are quality. Eventually, all quality can be represented through quantity. Properties qualitative and quantitative are just labels of information, our gathered indisputable fact.
If you graph any number system, eventually patterns appear. In mathematics, numbers and their patterns do not only continue infinitely linear, but in all directions. For example, considering infinite decimal expansion, even the shortest segments have an infinite amount of points.
Our universe and the numbers not only go on infinitely linear, but even it’s short segments have infinite points.
Why should this be? Why has Mother Nature found an evolutionary advantage in arranging plant structures in spiral shapes exhibiting the Fibonacci sequence?
We have no certain answer. In 1875, a mathematician named Wiesner provided a mathematical demonstration that the helical arrangement of leaves on a branch in Fibonacci proportions was an efficient way to gather a maximum amount of sunlight with a few leaves - he claimed, the best way. But recently, a Cornell University botanist named Karl Niklas decided to test this hypothesis in his laboratory; he discovered that almost any reasonable arrangement of leaves has the same sunlight-gathering capability. So we are still in the dark about light.
This proportion is the same as the proportions generated by successive entries in the Fibonacci sequence: 5:3, 8:5,13:8, and so on
As we go further out in the sequence, the proportions of adjacent terms begins to approach a fixed limiting value of 1.618034 . . . This is a very famous ratio with a long and honored history; the Golden Mean of Euclid and Aristotle, the divine proportion of Leonardo daVinci, considered the most beautiful and important of quantities. This number has more tantalizing properties than you can imagine.
The Golden Ratio/Divine Ratio or Golden Mean
The quotient of any Fibonacci number and it’s predecessor approaches Phi, represented as ϕ (1.618), the Golden ratio. The Golden Ratio is best understood geometrically by the golden rectangle. A rectangle unevenly divided resulting into one square and one rectangle, the square’s sides would have the ratio of 1:1, and the new rectangle would be exactly proportionate to the original rectangle – 1:1.618.
This iteration can continue both ways, infinitely. If you plot a quarter circle inside each of the squares as they reiterate, the golden spiral is formed. The golden spiral is possibly the most simple mathematic pattern that occurs in nature like shells of snails, sea shells, horns, flowers, plants. Numbers are only what we use to organize quantitative information.
The Golden Ratio can be applied to any number of geometric forms including circles, triangles, pyramids, prisms, and polygons. The golden ratio is formed by thirds within thirds, sixths, the connection between two and three, including every even and odd number itself. The ratio itself represents the transcendence of numbers, understanding our world is not numbers, but what numbers represent. Through the spiral, the ratio illustrates how the numbers, all quantities, are quality. Eventually, all quality can be represented through quantity. Properties qualitative and quantitative are just labels of information, our gathered indisputable fact.
If you graph any number system, eventually patterns appear. In mathematics, numbers and their patterns do not only continue infinitely linear, but in all directions. For example, considering infinite decimal expansion, even the shortest segments have an infinite amount of points.
Our universe and the numbers not only go on infinitely linear, but even it’s short segments have infinite points.
Why should this be? Why has Mother Nature found an evolutionary advantage in arranging plant structures in spiral shapes exhibiting the Fibonacci sequence?
We have no certain answer. In 1875, a mathematician named Wiesner provided a mathematical demonstration that the helical arrangement of leaves on a branch in Fibonacci proportions was an efficient way to gather a maximum amount of sunlight with a few leaves - he claimed, the best way. But recently, a Cornell University botanist named Karl Niklas decided to test this hypothesis in his laboratory; he discovered that almost any reasonable arrangement of leaves has the same sunlight-gathering capability. So we are still in the dark about light.
This proportion is the same as the proportions generated by successive entries in the Fibonacci sequence: 5:3, 8:5,13:8, and so on
As we go further out in the sequence, the proportions of adjacent terms begins to approach a fixed limiting value of 1.618034 . . . This is a very famous ratio with a long and honored history; the Golden Mean of Euclid and Aristotle, the divine proportion of Leonardo daVinci, considered the most beautiful and important of quantities. This number has more tantalizing properties than you can imagine.
