The Intriguing divine ratio of Nature

Bees and trees, hurricanes, and galaxy. What do they have in common? If I say fundamentally, they follow the same contour, will you believe? Yes, they are indeed the same. They are similar to many other structures; we find in nature. If you find something beautiful, it could be the magic of the golden ratio. It turns out, that a simple yet intriguing number has been concealed everywhere around us. Our beauties from Mona Lisa to Aishwarya Rai, they are all examples of the golden ratio. Am I exaggerating? Nope!

So, what is this golden ratio thing? Our golden ratio is a close neighbor of the Fibonacci series. We all must have come across them at least once in our lifetime. The Fibonacci Series is a set of numbers that increases rapidly, which began as a medieval math joke about how fast rabbits breed. But, it has much more to it. They are widely used in art, architecture and even nature make use of it in plants, animals as well as in other inanimate objects too. The terms of the Fibonacci series are obtained by summing the preceding two terms. The Fibonacci numbers are quirkier than you think they are.

For example, the difference between the square of a Fibonacci number and the Fibonacci number two terms before it, again yields a Fibonacci number.
5²-2²= 21 which is a Fibonacci number

The sum of any 10 consecutive terms of Fibonacci series is always divisible by 11.
2+3+5+8+13+21+34+55+89+144=374
which in turn divided by 11 gives 34

Also, dividing the consecutive terms of Fibonacci series, we get, 1/1=1, 2/1 =2, 3/2 =1.5, 5/3 = 1.667, 8/5= 1.6. Larger the terms, the closer it gets to the Golden ratio. Two numbers are said to be in golden ratio if the ratio of the sum of the numbers (a, b) divided by the larger number (a) is equal to the ratio of the larger number divided by the smaller number (a/b). The golden ratio is about 1.618…. and is represented by the Greek letter phi.

How Flora has it

Seeds in flowers, in every spiral, counts to a Fibonacci number. Most of the time, petals are also arranged in such a way. Sunflowers are the best example of Fibonacci in plants. No. of leaves in a turn also count to the golden ratio. The Fibonacci sequence is also found in the branches of trees. The number of branches at any time is mostly a Fibonacci number. Algae and fibrous roots also exhibit this trait.

Double Helical Beauty

DNA molecule measures 34 Armstrong by length and 21 Armstrong by width for each full cycle which are obviously Fibonacci numbers and their ratio when taken is 1.690476 which closely approximates Phi, 1.6180339. The length of our fingers, each section from the tip of the base to the wrist is larger than the previous one by about the ratio of phi.

How a Colony is Built on this

There are about 20,000 kinds of bees, and the honeybees are the most common ones. Even they are followers of math. Not just their hives inspired great scientists for the golden ellipse could be used as a geometric model for the spreading of light in optic crystals, their family tree too is a good example of the Fibonacci series. Wonder how?

Honeybees are social beings which live in hives. Every hive has a queen bee, few worker bees, and few male bees. When the queen bees are ready to mate, they mate with ten or more bees at a time. Because of this, few eggs go unfertilized and some are fertilized by more than one sperm. This causes male bees from the unfertilized eggs to have only one parent and the female honeybees(infertile) from the fertilized eggs may have more than two fathers.

Let’s look at the family tree of a male bee. A male bee has only one parent I.e. the queen bee. The queen bee has two parents, a male bee and a queen bee. Again, this queen bee has two parents, a male bee and a queen bee and the male bee has one mother which sums up to three grandparents. The queen bees have a mother and a father each, and the male bee has one mother which counts to five.  Notice the pattern. 1 male bee, 1 queen bee, 2 grandparents, 3 great-grandparents, 5 great-great-grandparents and goes on. 1,1,2,3, 5… Seen the sequence somewhere? Yeah, they are Fibonacci numbers. The ratio of male bees to the worker bees in any hive is always the golden ratio I.e. 1:1.618.

From DaVinci to Apple

Art and architecture widely use the golden ratio. The Greeks used the golden rectangle in The Parthenon. Even the pyramids have their own divine ratio. The ratio to the base and the slope of the pyramid is 1 to 1.618. Leonardo da Vinci used the golden ratio to paint Mona Lisa. Many photographs and arts also use golden ratio from the shades to the canvas. The Current logo that Apple Inc. uses has it too.

Golden Disks

The spiral galaxies also follow the Fibonacci pattern. What is the golden ratio in a spiral? Golden spirals are special cases of logarithmic spirals. Any radius vector which makes the same angle with the curve is called an equiangular spiral or a logarithmic spiral. The ratio of the distance from the center to each spiral arm and the adjacent is always constant in the case of a logarithmic spiral. But in the case of a golden spiral, for every 90°, the distance from the center of the spiral increases by the golden ratio i.e. 1.6180. That’s what makes it golden.

The Milky Way galaxy, our home has several spiral arms, each of them measuring a logarithmic spiral of about 12 degrees. An interesting fact is that spiral galaxies appear to defy Newtonian physics. As early as 1925, astronomers realized that, since the angular speed of rotation of the galactic disk varies with distance from the center, the radial arms should become curved as galaxies rotate. Eventually, after a few rotations, spiral arms should start to wind around a galaxy. But they don’t and hence the winding problem. The stars on the outside, it would seem to move at a velocity higher than expected — a unique trait of the cosmos that helps preserve its shape.

Nautilus shell is an example of the golden ratio in a spiral. The Cochlea of the inner ear, horns of certain goats, hurricanes even spider webs do follow them. Scientists believe a healthy vagina is associated with its dimension. It’s healthier when the ratio approximates the golden ratio.

Even a paper has been dedicated to the Fibonacci series and has been running since 1963 and according to its editor, they would not run out of articles any sooner since a lot of them are yet to be published.
Where there is beauty, there is a golden ratio! So, the next time when you look at something beautiful there is a good chance it has the number 1.61803398875…… and so on behind it. Where else have you observed this ratio? Feel free to use our comment box.

-Keerthana Vengatesan

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