Irrational Dimensions

[PerryAsbury]

Hey guys, tis my first entry so apologies if I seem a little nervous!

 

My friend and I were having a healthy debate on rather silly physics theories as we do. My friend then suggested the existance of irrational dimensions. This seemed far more silly than any other but i found some proof off a website i cant remeber the address of. I would like to open this thought up to debate - irrational dimensions or at least non-integer dimensions (like 4.3rd dimension or the e'th dimension)

 

Thanks all!!

 

Perry.

[[Tycho?]]

This is probably more of a math question than a phyics question. It might not make sense mathmatically, in which case it definately wouldn't be in physics. So I'd ask this over at the math forums if you dont get good responses here.

[timo]

The only time I have seen a non-integer value being associated with the term "dimension" was with the Hausdorff Dimension (http://en.wikipedia.org/wiki/Hausdorff_dimension). However, it´s more maths than physics and I can´t think of any relevance for relativity, atm. Are you sure the website you saw wasn´t a math-page?

 

EDIT: And I´d suggest that you don´t follow [Tycho]'s suggestion. I remember an incident not too long ago where someone in the same situation followed this advice only to be molested by someone else who complained why he started two threads with the same question ... .Just have the thread moved to the math section by a moderator if you feel it´s better suited there.

[Sayonara]

I'll move the thread, if desired...?

[guardian]

I seem to distinctly remember something I read recently about fractal dimensions and that they can be irrational. That may not be what you're after though. If it, is perhaps mathematics/geometry is a more appropriate section.

[PerryAsbury]

Thanks for suggestions of moving the thread. I will do soon and put a notice up telling people NOT TO MOLEST ME for doing so!!

 

Perry.

[Dave]

Yes, there are mathematical objects which posess the property of having a non-integer dimension; you've probably heard of them before because they're called fractals. In fact, one can think of a fractal as something which has a non-integer dimension (I'm sure others would disagree since there's no good description of what a fractal is, but this is beside the point).

 

A nice easy example of why one might consider using a non-integer dimension is given by the Koch snowflake (google for it and you'll find the appropriate construction). If you go ahead and measure the perimeter, [imath]p_n[/imath] at the n'th step, you'll notice that [imath]\lim_{n\to\infty} p_n = \infty[/imath]. But if you measure the area at the n'th step, [imath]A_n[/imath], [imath]\lim_{n\to\infty} A_n < \infty[/imath] (I can't remember the exact value off the top of my head). This would seem to imply that the Koch snowflake is somewhere in-between one and two dimensions.

 

Interestingly there's several ways of measuring the dimension of a set (more precisely, a metric space I believe). As Atheist mentions, there's the Hausdorff dimension. But there's also the box-counting dimension, Lyapunov dimension, and a number of other things as well.

 

It's a little confusing at first, but well worth putting the time in to study these objects. Fractals like the Mandelbrot Set are amazingly complex and yet are easily generated from something like a simple quadratic map. I hope this helps a little bit.

 

PS: If you want an "easy" fractal to get you started, take a look at the (middle third) Cantor set. It's just about the easiest you're likely to come across..