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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 But for sure they will lead to the area of fractals And regarding fractals, do not miss the new excellent fractal series
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
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.here on DA
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
********
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
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
A Christmas Greeting from Sweden
As a Christmas greeting from Sweden I will promote a YouTube channel by the nature filmer and photographer, Jonna Jinton. The link goes to her channel. Go and see the 15 minutes introduction video in full screen, the image below from the beginning of the video, The scenarios are a mix from both midwinter, midsummer, and other seasons. The soundtrack is as beautiful as the scenarios, and Jonna speaks in English with a relaxing voice. I myself live in the very south of Sweden about more than thousand kilometers south from the area where Jonna and her man and dog lives, the conditions of life being quite different. Here the sun shines for about 7 hours at midwinter. Just click the introduction video, put it in full view and enjoy :snowflake: :santa: :snowflake:
How I Became Interested in Fractals
This journal ought to have been written and published for at least a decade ago The year 1987 it was an India festival in Sweden. At that time, besides my work (I was a shift worker), I was studying Sanskrit at the university of Gothenburg. My special interest was the way they wrote grammar in ancient India about 500 BC. An evening at the late autumn I was sitting with some friends in Stockholm, the capital of Sweden, and watched a TV program about Indian dance. At the end of the program, the speaker talked a little of the theory behind the dance. I don't remember what he said in details, but it was something about how the god of dance, Shiva, transformed order into chaos and chaos into order. Then a map was shown of the coast line of Great Britain and the speaker quite naturally went into that "today the mathematicians has found that there is an order behind the irregular". The subject was the new science about chaos and complexity. Both Mandelbrot and
A special flashmob
This is journal I ought to have written and published for at least one year ago. But as we say in Sweden, ”better late then never”. So first, what is a flashmob? ”A flash mob (or flashmob) is a group of people who assemble suddenly in a public place, perform for a brief time, then quickly disperse, often for the purposes of entertainment, satire, and artistic expression. Flash mobs may be organized via telecommunications, social media, or viral emails.” Quotation from Wikipedia. Back to August, 2021, when there still was some Covid restrictions in Sweden, a group around the Swedish musician and producer Christoffer Lundquist started to perform some flashmobs at some squares in Malmo (the third biggest city in Sweden) and Lund (Sweden's Oxford) at the very south of Sweden. He had just produced a new song that was a protest against the Covid restrictions we had at that time. Now remember that those restrictions were the most liberal in the western world. Yet we were less affected by
The Dreams of World Economic Forum
Hi everyone, Yuval Noah Harari is the one of the advisors to the founder and chair man of World Economic Forumm (WEF) Klaus Schwab.You can read about him (Harari) at Wikipedia.There is an interesting channel at BITCHUTE, HighImpactFlixwho has uploaded 2 videos, 9 minutes and 14 minutes where Harari dreams trans humanistic dreams, Something REALLY BAD is About to Fundamentally Change Humanity - (Pt 1) Some quotes from the video: "What should we do with all these useless people" (Already discussed, due to the fact that about 90 percent of the today's employments will disappear as a consequence of the fourth industrial revolution.) "People could look back in a hundred years and identify the corona epidemic as a moment when a new regime of surveillance took over, especially surveillance under the skin". (You probably know that you may have the vaccine passport as a chip under your skin. The proposals of vaccine passorts where already there in EU as a ”health passport”. Social
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Thanks for making this journal! It is very interesting and helpful!