Golden Ratio

As defined by Wikipedia, “Two quantities are in the (state of the) golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller (one).” This may be a difficult concept to grasp, so let’s use a picture.

A is to B

Lets assume that the segment a+b = c. The ratio of c to a (or c/a) is equal to the ratio of a to b (or a/b).

If the ratio is equal to Phi, is it said to be the Golden Ratio. So, for simplicity, the Golden Ratio is equal to Phi (1.6180339887…) But why Phi?

The Golden Ratio is everywhere in our natural world and shows up unexpectedly in the strangest places. See the different applications.

As you probably know by now, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, etc.

Let’s divide the first two values of this amazing sequence.

1/1 = 1

The next two…

2/1 = 2

 

The next two…

3/2 = 1.5

 

And so on…

5/3 = 1.666666

8/5 = 1.6

13/8 = 1.625

As you can probably see, the farther on you divide, the closer the ratio gets to the Golden Ratio. This means that no ratio of two Fibonacci numbers can equal the Golden Ratio. This also means that the farther in the sequence you divide with the values of the Fibonacci sequence, the closer and closer you get to the Golden Ratio, but you will never reach its exact value.

The Golden Ratio is an example of a limit. The amount that you are off by is known as the error.

 

The Fibonacci sequence has many other applications.


Spiral

 

As you can see in the image above, starting with the green 1 X 1 square, we add the previous dimensions to get the next one.

  1. 1 X 1
  2. 1 X 1
  3. 2 X 2
  4. 3 X 3
  5. 5 X 5
  6. 8 X 8
  7. 13 X 13

This follows the Fibonacci sequence with a geometric take. Literally adding lengths of each side to create the next, it is easy to visualize this concept. Dividing the previous square gives you the number approximation for Phi. The more squares you create, the more accurate approximation you make to Phi.


The rectangles as a whole are known as golden rectangles.  This uniquely designed rectangle has the ratios of (Width to Length) 1 to Phi (1.618…). As understood by above, the farther you keep going out, the closer the dimensions get to Phi.

This rectangle is known to be pleasing to the eye. Its properties have been used in various art.