non integer dimensions

Higher-dimensional geometry (previously "Polyshapes").

non integer dimensions

Postby catwoman » Tue Oct 26, 2004 4:11 pm

well I think that's what you call one and a half D etc.
I'd forgotten all about this which I learnt when looking at fractals.
Does anyone here have interesting information or links to sites ( apart from cynthia lanius' fractal site) that are about the non integer dimensions?
cheers,
Carolyn
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Postby houserichichi » Thu Oct 28, 2004 6:13 pm

Well if you're looking for something other than basic "in words" explanations, try

http://en.wikipedia.org/wiki/Hausdorff_dimension
http://www.geisswerks.com/ryan/NEAT/FAQ/fractal1.htm

and if you'd prefer something a little less mathy, try

http://www.crca.ucsd.edu/~syadegar/Mast ... ode31.html
http://www.jracademy.com/~jtucek/math/dimen.html

and if you feel so inclined, look up "fractional dimensions", "non-integral dimensions", "fractal dimensions", and any other combinations you can look up on google. Also, if you're really keen in learning about things, try just about any book on chaos or fractals. There are a plethora of introductory books out there that are pretty light on the math (look for them on Amazon or Chapters...the more popular books tend to be the easier reads but the least informative). Hope that helps a little.
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Postby Paul » Tue Nov 02, 2004 6:35 pm

Hello all,

Perhaps this is a nonsensical question...

I'm wondering does it make any sense to think that say, a non-integral rational dimension can essentially be converted into an 'essentially similar' integral dimension simply by multiplying through by the denominator of the non-integral rational dimension?

If yes, in what sense(s) might two such dimensions be 'essentially similar'?
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Postby pat » Thu Nov 04, 2004 6:22 am

Paul wrote:Perhaps this is a nonsensical question...


I'm trying to think about how this might work. But, I keep running into the Strong Law of Small Numbers. I was trying to think about what one would have to do with say, dimension a*b where both a and b are integers. But, I have to get up to dimension 2*3 to be useful at all since: 1*n = n and 2*2 = 2+2.

Anyhow, my first thought was to take the Cartesian product of the spaces. So, if (x,y) is one space and (z) is another space, then (x,y,z) is in the new space. Unfortunately, if the original spaces are of dimension a and b, the new space will be of dimension a+b rather than a*b.

And, taking all of the linear functions from a space of dimension a to a space of dimension b results in a space of dimension b<sup>a</sup> (if I'm thinking about that right).

So, I need something between addition and exponentiation here.

Of course, the formulas for fractal dimensions involve logarithms, so maybe exponentiation is a good way out.

But, as for some of the side questions in the meantime.... 'similar' would probably have to mean homeomorphic while 'essentially similar' would mean that the homeomorphism is "natural" in some way. (Sorry, I couldn't find a good link for 'natural'.)
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Postby Paul » Mon Nov 08, 2004 8:43 pm

Hello Pat,

Well, it sounds like it's pretty much a nonsensical question.

I don't know much about non-integral dimensions.
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Postby RQ » Wed Nov 10, 2004 12:30 am

Aale de Winkel wrote:fractional "dimensions" you have in figures call fractals this way:
the figure _/\_ is a line having 4 parts in the "space" of 3 parts and thus has "dimension" 3/4
this kind of dimension has nothing to do with the dimension of a space.
to distinguish one probably best call this "fractal dimension".
as a nutural example a shore-line is a sample of a line having broken "fractal dimension".

there are various "fractal generating programs" around.
the fun with fractals is that one can zoom in on some part and essentially end up with the same picture, consider pe each of the four lines above also having the same _/\_ shape etc ad infinitum.
with shorelines, snowflakes etc the same thing happens.

see http://mathworld.wolfram.com/Fractal.html for various samples.
so no-one is wrong, it is simply quite something else
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