The Beauty of Nature

"There once was an Ugly Duckling, whose..."

As Danny Kaye would have said.

Nature says different.

Has anyone ever wondered why a rectangle is more pleasing to see than a square?

Or that doors are rectangular rather than any other shape?

Letterboxes are not not circular?

Sidewalk slabs are not round?

Bricks are brick-shaped?

There is a natural number OTHER than "e", that figures in all of this. It's name is "Phi", (yawn), yes another Greek letter. It's value is about 1.618033988.

It is the only number that becomes it's own reciprocal by subtracting unity. (to those unfamiliar with these terms, let me just say that 0.618033988 is the same number as 1/1.618033988)

This writer's first encounter was when she used arithmetic series and geometric series. The first is where each number is the result of adding another number to the same one all the time, for example 1,3,5,7,9,11,13 thus adding two to each. The second is where you multiply instead, 1,3,9,27,81,243,729 etc multiplying them all by three. I chose the first two numbers 1 and 3 as an example.

The first is governed by a "common difference", the second by a "common ratio". The first has common difference of 2, the second, a common ratio of 3.

My question to myself when I was 13 (and a half) was "What is the algebra for the case when "common ratio" is the same as "common difference" ?"

The answer was a simple quadratic equation, the solutions of which were 1.618033988..... and 0.618033988...



There once was an Ugly duckl.... no, there once was a guy named Fibonacci, whose numbers were stuffy and frowned (upon) .

They were...

1, 3, 5, 8, 13, 21, 34 55, 88.... etc

Get outta town!!

Every number was the sum of the two before.

I advise you to get a calculator out at this point and point it at two numbers that are side-by-side in that sequence, no, no! , continue the sequence by adding the last two together. Divide them. You will will either get 1.618033988... or 0.618033988... depending on which was the denominator.

The "Fibonacci Sequence" obeys the same rules as the "Common" differences and ratios as mentioned previously.

It is also present in sunflowers, the way trees branch, fractals, and all sundry items nature intended us to find it in.

I would advise the reader to go check out "Golden rectangles" and logarithmic spirals.

What happens when you intersect three golden rectangles in the three planes?