Whirling Squares Spiral

Construction of the Whirling Squares' Spiral

The light grey squares are marked off within this rectangle. They form what are called the whirling squares. Note that in a golden rectangle with aspect ratio 1.618... a smaller golden rectangle is formed with the part left over after constructing the square.  And this continues as we 'whirl' around.

https://en.wikipedia.org/wiki/Fibonacci_sequence:

In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

Whirling Squares (with 'eye' at lower right)

My paintings often have Fibonacci number dimensions ; eg, 5x8", 8x13" and multiples of the 5x8: 10x16" and 15x24" (near-PHI paintings, where PHI is the golden mean - 1.618...). 13x21 and even 21x34 are dimensions to be exploited in the future.

For these (near) PHI paintings I like to begin my design with a golden spiral superimposed on a dynamic symmetry armature, taking advantage of the benefits of both diagrams:


The blue spiral is called a whirling squares' spiral, formed by quarter circles within the squares constructed within this golden rectangle. This spiral approximates the mathematical definition of the Golden Spiral, which is a logarithmic spiral with growth factor PHI. An excellent resource showing the relationship between a Golden and the Whirling Squares' spiral is https://en.wikipedia.org/wiki/Golden_spiral#Approximations_of_the_golden_spiral.  For more information on the Golden Mean visit the Bibliography post in this blog.

One especially informative image is: 

https://en.wikipedia.org/wiki/Golden_spiral#/media/File:GoldenSpiralLogarithmic_color_in.gif. This is a dynamic 'gif', where you can see the spirals moving.

Here's a snapshot:




This diagram shows how close the Fibonacci Whirling Squares' Spiral is to the Golden Spiral!:



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