In the beginning mathematics was based on numbers which we all use every day to add, subtract, multiply and divide. Next we developed symbols such as ( x, y, z ) to put into formulas to solve for numbers. Later on we developed trigonometry which is essentially the study of ratios involving geometry and then came calculus which was the study of what happens when you continually shrink / change distances until you get something very small. Somewhere along the line, people started to put ideas into mathematical form and solved for an outcome which was later proved / disproved in experiments. Einstein's Theory of Relativity is an example of that phenomena. Mandelbrot discovered fractals using equations which resulted into some very beautiful designs. The discovery of fractals also revealed something called an attractor which was a number, around which these beautiful fractal designs seem to evolve. If you live long enough, most of us will realize at some point or another that most of the time there is stability in our lives, then sometimes instability and finally outright chaos. Fortunately, cycling also exists, so with a little luck we all survive without too much damage. When you think about it, there is a possibility that all seemingly chaotic systems retain some shreds of order. Usually statistics is used to find these correlations of order which drives most people batty. Maybe there is a simpler way. If you take the numbers from 1 to infinity and add their digits, you will find that the one digit totals will be one of ( 1, 2, 3, 4, 5, 6,7, 8, 9 ) in sequence. These one digit numbers are the fractal attractors of our numbering system.
For instance, the number ( 97 ) has the digits ( 9 and 7 ). Add the digits ( 9 + 7 = 16 ). Keep adding until you have a one digit total ( 1 + 6 = 7 ). Number ( 97 ) has the number ( 7 ) as its' attractor.
If you graph the one digit attractors of all the numbers from one to infinity you will have a series of even right angle triangles ( _!, _!, _! , etc. ) that look like waves or the teeth of a hand saw which in essence form a fractal.
If you subtract the one digit total attractor ( 7 ) from the number ( 97 ), ( 97 – 7 = 90 ) and graph the results for all the numbers you will get a series of elongated climbing steps which is really a series of butted rectangles forming a fractal in the shape of an elongated staircase.
If you divide ( 90 ) by 9 ( 90 / 9 = 10 ) and do the same to all the other numbers and then graph you will also get a series of elongated climbing steps which is, once again, a series of butted rectangles forming an elongated staircase.
The single digit numbers are the attractors of our numbering system. The single digit numbers repeat themselves in an ordered pattern, from 1 to 9 and then repeat 1 to 9 again and again. You will see from the graph that the 1 digit numbers form a series of uniform right angled waves ( _!, _!, _!, etc. ) which is an infinite saw toothed fractal.
This is the model for the quantization of any number or set / system of numbers whether sequential, harmonic, energy related, chaotic, decimal, fractional or otherwise into attractors and fractals. The only limitation is your imagination. You then graph the attractors or rectangles which form into waves, steps, mountains or, in general, fractals.