posted
George Cantor was a 19th Century Mathematician, and, obviously, is credited with the development of the Cantor set.
The Cantor set is, in essence, a completely empty space, just more interesting. It is constructed by recursively removing thirds.
Take a line of any length. Now take out the middle third of the line:
code:
Original: ----------------------------------------------- One Iteration: ----------------- ----------------
You are left with two lines, each a third the length of the original line, separated by a space equal to a third of the line. Now take each of these two lines and pretend they were the original. In other words, perform exactly that same removal on each of them.
code:
Original: ----------------------------------------------- One Iteration: ----------------- ---------------- Two Iterations: ------ ------ ------ ------
You now have four lines, each one separated by a specific spacing. Then take each of the four lines and remove the middle third.
code:
Original: ----------------------------------------------- One Iteration: ----------------- ---------------- Two Iterations: ------ ------ ------ ------ Three Iterations: -- -- -- -- -- -- -- --
Now you have 8 lines, repeat the process on those 8 lines, and on the 16 lines you’d get then and on and on into infinity.
This is a Cantor set, it’s shown up in the natural world. For instance, transmission of information over electrical wires has noise in it, when seen as a data set, the noise follows the Cantor set perfectly, helping electrical engineers model the disruptions that are a natural result of electrical transmission.
In the above mentioned case, the amount of electrical disturbance at any time period can be measured by adding up the number of lines within that period. At some points there is absolutely no disturbance at all. At others, disturbance comes in very high doses. Yet even in the highest disturbance periods, there are periods within it that have no disturbance. No matter how finely you break it down you will never find any length of time without some disturbance. So how does this relate to Hatrack? Political threads, cycles. Sometimes Hatrack is almost complete fluff, no contention, just general discussions. At other times there is violent discussion, arguments that rage over the board and the moderators have to be called in and people banned. Yet even when these threads seem like they’re taking over the board, there’s always some fluff threads, in fact the boards are always full of them.
So I personally think the boards political discussions can clearly be modeled by a Cantor set. Neat, huh?
[By the way, I've tried to get the [code*] ubb command to show those lines better with the preview post... but I couldn't. Sorry.]
posted
I've never heard of cantor sets before, but I liked the explanation. I'm not entirely clear on the electrical noise example though. Are you merely saying that such a phenomenon is periodic, or is there more to it than that?
Posts: 4548 | Registered: May 2001
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posted
It's periodic, and there's also the phenomenan that over any span of time there will be periods of time with no noise. Even when there are huge spikes of noise, there will be periods within those spikes that are noise free.
posted
I wonder if there isn't a relationship between cantor sets and Fourier series. When applied to electrical signals, at least, it seems that the two may be alternative models of the same phenomenon.
posted
Ahh the summation would occur in expressing the size and number of the gaps.
You would have to go
1/3x*2^0 + 1/3(1/3x)*2^1+ 1/3(1/3(1/3x))*2^2
So the size of all the gaps together would be expressed as
Sum [((1/3)^n)x*2^(n-1)]
The number of gaps would be expressed as 1/2[((1/3)^n)x*2^n] or ((1/3)^n)x*2^(n-1)
The size of the smallest gap at each iteration would be the same size as the smallest line segment at each iteration which would be the same as what I stated in the previous post.
Posts: 11265 | Registered: Mar 2002
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posted
It strikes me that there’s something wrong with your math AJ, at least for total number of gaps.
code:
One Iteration: 1 gap Two Iterations: (1+2) 3 gaps Three Iterations: (1+2+4) 7 gaps Four Iterations: (1+2+4+8) 15 gaps Five Iterations: (1+2+4+8+16) 31 gaps
So it’s simply:
code:
2^n – 1
Neither x nor the 1/3 have anything to do with number of gaps.
posted
As someone who did her senior paper on fractals (heavily featuring the cantor set), I am delighted with and applaud this comparison. Posts: 1805 | Registered: Jun 1999
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quote: In his work Fourier was guided as much by his sound grasp of physical principles as by purely mathematical considerations. His motto was: “Profound study of nature is the most fertile source of mathematical discoveries.” This brought him biting criticism from such purists as Lagrange, Poisson, and Biot, who attacked his “lack of rigor”; one suspects, however, that political motivations and personal rivalry played a role as well. Ironically, Fourier’s work in mathematical physics would later lead to one of the purest of all mathematical creations—Cantor’s set theory.
posted
Quite possible AJ, I just saw the equation who had for number of gaps and realizied that x had no place it in, kind of a tip off.
Rivka, I'm not sure what Cantor's set theory is really, but it's a theory, where this is just a Cantor set. Kind of like (1,2,4,8) is doubling set, and the theory "in a set starting from some base point will go to infinity if the absolute of the base is greater than 1 and the set doubles" is a doubling theory. I have no clue how connected the Cantor set is to Cantor's set theory though.
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