Mathematical Playtime 1

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With this journal the “Mathematical Playtime” series is started by me ;) It will not be a long series, only three or four journals I think ;) But who knows :hmm: But for sure they will lead to the area of fractals :) And regarding fractals, do not miss the new excellent fractal series All About Fractals Part 1 -- What Are Fractals?This is the first part of a series of articles about fractals, aiming to introduce the mathematical side of fractals to the DA community, who are familiar with the artistic aspects of them. The information given will be very basic and won't require anything beyond basic mathematical knowledge.
Part 1: What Are Fractals? : An Introduction
Part 2: The History of Fractals - 1 : Fractals before Gaston Julia
Part 3: The History of Fractals - 2 : Julia and Mandelbrot Sets
Part 4: Fractals and Computers : IFS and Escape-Time Fractals
Part 5: Fractals and Art : Mathematics Meet Art
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What Are Fractals – An Introduction
This question is one of the killer questions you can ask a fractal artist, because the answer is likely to be either too complicated or not explanatory enough. So, in order to help the artists remain sane, read this article for the answer to that question. :)
The American Heritage Dictionary defines “fractal” as “a geometric pat
here on DA :dance:

Now we will play the game I am a rational number between 0 and 1. Double me time after time, but as soon I become 1 or greater, subtract 1, and go on for ever.

Complicated? Well let’s take it more easy ;)
First:  What is a rational number? Answer: A rational number is a number that can be brought into the form p/q where both p and q are integers.
Second:   We can put our game into the iterative rule:

p/q  ->            
2(p/q) if 2(p/q) still being less than 1
2(p/q) – 1 if 2(p/q) being 1 or greater


Still complicated? With some examples I think no :) So let’s start:

1/6 -> 2(1/6) = 2/6 = 1/3 -> 2/3 -> 4/3 – 1 = 4/3 – 3/3 = 1/3. Arriving back to 1/3 (1/6 -> 1/3 -> 2/3 -> 1/3) we have been caught into a 2-periodic cycle (1/6, 1/3, 2/3), the underlining is meant to mark a repeating cycle.

Yet still Complicated? Well all this can be seen more obviously if we regard our rational numbers (in fact all real numbers) between 0 and 1 as fractions of a whole turn. Then 1/6 corresponds to 360/6 degrees = 60 degrees, 1/3 to 120 degrees, 2/3 to 240 degrees and 1 to 360 degrees. When we double 2/3 (240 degrees) to 4/3 (480 degrees) we in fact have returned to 480 – 360 degrees = 120 degrees, that is 1/3 of a whole turn.

From the above we can already make the statement:
If we start with a rational number with an odd denominator we are situated in a periodic cycle. If we start with a rational number with an even denominator we after some iterations (in fact when the denominator becomes odd after some abbreviations) will be caught into a periodic cycle, that is they are preperiodic.

Let’s have some more examples:
1/7 -> 2/7 -> 4/7 -> 8/7 - 1 = 8/7  - 7/7 = 1/7. In other words, 7 in the denominator gives rise to a 3-periodic cycle. However we also have 3//7, 5/7 and 6/7. So if we start with 3/7, we obtain the orbit; 3//7 -> 6/7 -> 12/7 – 1 = 12/7 – 7/7 = 5/7 -> 10/7 - 1 = 10/7 - 7/7 = 3/7. In other words, period 3 has 2 cycles:

(1/7, 2/7, 4/7), and back to 1/7.
(3/7, 6/7, 5/7), and back to 3/7.

Above we have dealt with rational numbers that have the odd numbers 3 and 7 as denominator. What about 5 as denominator? Let’s test : 1/5 -> 2/5 -> 4/5 -> 3/5 and back to 1/5, in other words a 4-periodic cycle. However there are more rational numbers that have 4-periodic cycles. In order to include all these we must write our rational numbers in a form that has 15 as denominator. Thus:

(1/15, 2/15, 4/15, 8/15),  and back to 1/15.
(3/15, 6/15, 12/15, 9/15), and back to 3/15.
(7/15, 14/15, 13/15, 11/15), and back to 7/15.

The second cycle (3/15 etc) in fact is the same as 1/5 etc as we were looking for first. Note that 5/15 etc is not of period 4 as 5/15 = 1/3 and that rational is 2-periodic. The same is of cause true for 10/15 = 2/3. In fact if you want to check which denominator “q” you must have to include all cycles of a period “ per” the rule runs q = 2^per – 1. So for period 2 the denominator is 2^2 – 1 = 4 – 1 = 3, for period 4 the denominator is 2^3 – 1 = 2*2*2 – 1 = 7, for period 2 the denominator is 2^4 – 1 = 15 etc as we have seen above.

This will be enough for this journal. In the next journal in the Mathematical Playtime series we will, believe it or not, learn about an even more easy method to find periodic cycles. Until that, take care and don’t become doubled beyond one :dance:
© 2007 - 2024 FractalMonster
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Thunderheart1287's avatar
Thanks for making this journal! It is very interesting and helpful! Clap