NON-INTEGER Dimension

hey!

i'm writing a research paper (extended essay) in maths on dimension and my research question is about exploring its different facets. to do this i'm talking bout fractals, fractal geometry, fractal dimension, deterministic chaos, and the moebius strip & klein bottle

right now i'm stuck on fractal dimensions :s-smilie:

, or rather dimensions that are not an integer - not 0D, 1D, 2D, 3D, 4D, ... the typical stuff we learn in junior school

WHAT IS, and how can i explain, the significance and meaning of non-integer dimension??

THANK YOU!!!

Reply 1

Alive

http://en.wikipedia.org/wiki/Fractal_dimension

??

I don't think you should think about it in comparison to 2D, 3D. After all, what does 6D really mean?

Reply 2

Adalia

Alive

http://en.wikipedia.org/wiki/Fractal_dimension

??

I don't think you should think about it in comparison to 2D, 3D. After all, what does 6D really mean?

thanks but that doesn't really help me - i know what it is, and how to calculate it, but i don't know what it MEANS* in this world... does somebody here?
or know a site that could help?

oh and sorry if that came across wrong, i'm not comparing to 2D, 3D (was just clarifying for the reader)

* = not this (from that wiki page): " how completely a fractal appears to fill space"

--> basically, what is the MEANING of an object with dimension 1.3 ... what is it, what does that MEAN for us, (how does one explain it)?

thank you :smile:

Reply 3

generalebriety

What exactly are you asking? There's a formal definition there. Just because they've called it "dimension", doesn't mean you should expect it to correspond to what you think is meant by "3-dimensional" (though of course there's a relation). If they called it something else, would you be asking this question?

I'm not trying to rubbish your project here and be awkward, I just don't know what you want to know.

Reply 4

Adalia

generalebriety

What exactly are you asking?

that! ==>

Adalia

basically, what is the MEANING of an object with dimension 1.3 ... what is it, what does that MEAN for us, (how does one explain it)?

but more on the MEANING part of it
(does that make sense?)

and nono, no worries

Reply 5

generalebriety

Adalia

that! ==>
but more on the MEANING part of it
(does that make sense?)

and nono, no worries

The meaning of "a fractal of dimension 1.3" is precisely what the article suggests: the limit as epsilon --> 0 of log N(epsilon) / log(1/epsilon) is 1.3. The article itself does say, though, that this is only one of many different definitions of 'dimension' and that the concept isn't particularly well defined - and, in particular, that a given fractal might have two different 'dimensions' using different definitions of the word. And - though I'm not an expert - I don't really think this has very much to do with things like "1-dimensional space" or "3 dimensions". I don't think the two meanings of the word are very strongly correlated.

You might want to look up "Hausdorff dimension", which is a related concept.

Reply 6

Adalia

thanks

, i've gotten all that, but i was wondering what a dimension of 1.3 means, like what it is
(analogy: something 1D is a line)

maybe its a stupid question?

Reply 7

Alive

What does 6D mean? What does 3D mean? What does the number '5' mean?

You want to think of things in terms of what you already know, but that isn't really applicable. In maths, even when we have nice ways to think about things, we still really mean what the definitions say - the ways of thinking are just nice little lies to help get your head around things.

For example, it might be nice to say 0.9(recurring) = one, but what we really mean is that the limit of the sum of 9/(10^j) tends to 1 as j tends towards infinity.

And what we mean by that is the limit of the partial sums blah blah blah epsilon >0 etc.

And to actually answer your question "How does one explain it?" - You explain as in the wiki article, and if the people you are explaining it to don't understand, they are out of their depth.

Sorry if that came across as condescending, it wasn't meant to be.