Fibonacci series and golden ratio unite mathematics with the wonders of natures.Let it be flowers or monuments or even beautiful painting they have some relation with this ratio which makes mathematics much more beautiful.
If we consider the ratio of two adjacent Fibonacci numbers, it converges to a fixed number as it approaches infinity, this fixed number is known as the golden ratio.
For those who are unaware of it; Golden ration is mathematically defined as
a...b....a+b.... is a sequence, then if a+b/a(or b) = a/b, then this ratio is termed as golden ratio this happens to be equal to a(n+1)/a(n) as n approaches infinty. If you solve the quadratic equation we obtain the value of golden ratio = 1.618033.
Now here comes the magic of this ratio:
The number of turns between each newly grown cell of sunflower requires 0.618 turns to have a gap-less model to utilize a major proportion of daylight.
The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory's 21, the daisy's 34, and so on. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors.
In some cases, the seed heads are so tightly packed that total number can get quite high — as many as 144 or more. And when counting these spirals, the total tends to match a Fibonacci number.
Similarly, the seed pods on a Pine-cone are arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a 3-5 cone is a cone which meets at the back after three steps along the left spiral, and five steps along the right.
Fruits and Vegetables:
Likewise, similar spiraling patterns can be found on pineapples and cauliflower.
Human faces:
Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).
It's worth noting that every person's body is different, but that averages across populations tend towards phi. It has also been said that the more closely our proportions adhere to phi, the more "attractive" those traits are perceived. As an example, the most "beautiful" smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. It's quite possible that, from an evo-psych perspective, that we are primed to like physical forms that adhere to the golden ratio — a potential indicator of reproductive fitness and health.
Read more from this link 15 Uncanny Examples of the Golden Ratio in Nature
And there are certain applications which rates your attractiveness by predefining 100% as the golden ratio face.
Architecture:
The great pyramids of Giza:
The largest of the pyramids in Giza contains the use of phi and the golden ratio. The golden ratio is represented as the ratio of the length/height of the triangular face to half the length of the square base.
The length of the base of the pyramid is approximately 1+1 = 2
The height of an isosceles triangular face is approximately phi
The height of the pyramid is approximately the square root of phi
The height can then be found as
The slope of the pyramid is very close to the golden pyramid inclination of 51° 50’.
The following diagram may provide a more clear representation of the golden ratio in the façade of the Parthenon:
In the above figure, one can see the use of a golden rectangle that is Phi times as wide as the height of the structure. Notice the rectangles constructed and the highlighted portions of the segments. Each segment length of the rectangle follows the golden ratio, where the ratio of the lengths of the smaller yellow segment to the larger blue segment is equal to the ratio of the lengths of the blue segment to the whole white segment.
The golden ratio can also be found throughout the floor plan of the Parthenon:
The floor plan area is a
rectangle: the length is
times as long as the width of the ancient temple. Despite the numerous mathematical occurrence of the golden ratio in the construction of the Parthenon, there are no historical records of the original plan of the temple.
THE HEAD QUARTERS OF U.N:
The more obvious application of the United Nations headquarters to the golden ratio is found when looking at the width of the entire building and comparing it to the height of every ten floors. Refer to this link for more Page on uga.edu
The DNA:
So, the ratio about which you are reading right now is already in your genes..
Golden ratios from sides of canvas
So, the painting of Mona Lisa has traces of golden ratio (the most beautiful and perfect face as stated by critics)
- In the distance from the Da Vinci’s guideline drawn at the hairline to the guideline at the foot, the following are all at golden ratio points:
- the navel, which is most often associated with the golden ratio of the total height and not the height of the hairline
- the guidelines for the pectoral nipples
- the guidelines for the collar bone
- In the distance from the Da Vinci’s guideline drawn at the elbow to the guideline at the fingertips
- the base of the hand is at a golden ratio point.
Credits : Divine Proportion/Golden Ratio in the Art of Da Vinci
Similarly this half eaten apple :P logo of apple inc can be associated with the golden ratio.
HONEY BEES
Both the Fibonacci numbers and the Golden Ratio appear in Honeybees.
The Fibonacci numbers are very well represented in the honeybees. For example, the if you follow the family tree of honeybees, it follows the fibonacci sequence perfectly. If you choose any hive and follow this pattern, it would look like this:
If you divide the number of female bees by the number of males bees you get 1.618, the golden ratio. This mathematical sequence work for any honeybee hive at any give time. Commonly, honeybee hives are always used to explain the Fibonacci Sequence and Golden Ratio
And the list never ends, so all this is associated with the Fibonacci series, but I think, Fibonacci just rediscovered the sequence which was already known since ages.. Not just by humans but also by plants, insects and even our body cells... just trying to be sarcastic...