This is topic A short and silly Cousin Hobbes: The Cantor set explanation of Hatrack in forum Books, Films, Food and Culture at Hatrack River Forum.


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Posted by Hobbes (Member # 433) on :
 
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:

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Original:           -----------------------------------------------
One Iteration:   -----------------                   ----------------

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code:

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Original:           -----------------------------------------------
One Iteration:   -----------------                   ----------------
Two Iterations: ------      ------                   ------      ------

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code:

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Original:             -----------------------------------------------
One Iteration:     -----------------                   ----------------
Two Iterations:   ------      ------                   ------      ------
Three Iterations: --  --       --  --                    --  --      --  --

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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.]

Hobbes
 

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Posted by Richard Berg (Member # 133) on :
 
It probably helps to compose your post in a fixed-width editor, then use {code}.

Cantor sets are cool.
 

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Posted by Jacare Sorridente (Member # 1906) on :
 
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?
 

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Posted by Hobbes (Member # 433) on :
 
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.

Hobbes
 

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Posted by Annie (Member # 295) on :
 
Did I mention how cute and adorable you are when you use science metaphors?

Also - I'm impressed you're up so early. Have fun on the field trip!
 

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Posted by Jacare Sorridente (Member # 1906) on :
 
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.

[ March 26, 2004, 08:44 AM: Message edited by: Jacare Sorridente ]
 

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Posted by Scott R (Member # 567) on :
 
:scratches head:

Uh, okay.
 

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Posted by Hobbes (Member # 433) on :
 
First: Annie: I had a very enjoyable time, the whole thing was basically getting into Jeeps and 4-wheeling over hills and water and mud pits.

Scott, was that for me or Jacare... or both?

Jacare, I'm not sure what connection you're seeing there, or at least I can't make that connection myself and now I'm curious. Can you explain more?

Hobbes
 

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Posted by BannaOj (Member # 3206) on :
 
Let's see

so starting with X

next you get (1/3 x)*2

Then you would get 1/3(1/3x*2)*2

1/3(1/3(1/3x*2)*2)*2)

So the general equation for the iteration is

((1/3)^n)x*2^n

where ((1/3)^n)x is the size of the smallest line at N iterations and 2^n gives you the number of actual line sements that exist at that interation.

Does look suspiciously like a reduced fourier series but I can't remember the actual fourier series general formula.

AJ
 

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Posted by BannaOj (Member # 3206) on :
 
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.
 

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Posted by Dan_raven (Member # 3383) on :
 

quote:

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I personally think the boards political discussions can clearly be modeled by a Cantor set.

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code:

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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

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code:

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2^n – 1

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Neither x nor the 1/3 have anything to do with number of gaps.

Hobbes

[ March 26, 2004, 03:15 PM: Message edited by: Hobbes ]
 

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Posted by xnera (Member # 187) on :
 
As someone who did her senior paper on fractals (heavily featuring the cantor set), I am delighted with and applaud this comparison.
 

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Posted by rivka (Member # 4859) on :
 

quote:

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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.

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Posted by BannaOj (Member # 3206) on :
 
ok I made it way to hard as is my usual wont. I stand corrected on the number of gaps... but I think I'm right about the sizes.

AJ
 

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Posted by Hobbes (Member # 433) on :
 
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.

Hobbes
 

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