From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: What about 3.4 dimension?
Date: 10 Sep 92 23:50:19 GMT
Organization: U of Cambridge, England

In article <Sep10.194827.29156@yuma.ACNS.ColoState.EDU>, be231642@longs.LANCE.ColoState.EDU (Bret Egan) writes:

> Is there any literature about the possibilites of dimensions of real numbers?
> 
> R^1.2 or R^pi

Well, there are certainly notions of dimension that can give non-integral
answers. It's not a case of fractional-dimension SPACES, though (I really
don't see any chance of R^pi making sense), but of fractional-dimension
SETS.
This sort of thing is all tied up with fractals, so I suppose it's quite
trendy at the moment. Here's one way of looking at it...

One way to figure out the dimension of something is to try covering it with
lots of little balls or boxes or whatever, and see how the number of balls,
boxes or whatever needed varies as the <whatever> get smaller.

So if you've got a 2-dimensional thing, halving the size of your bits means
you have to multiply the number of them by 4; but if you've got a 1-dimensional
thing, you can sort of string the little balls/boxes/whatever along it, and
you only need about twice as many when you halve the size.

 (If you don't see this: Draw on a piece of paper (i) a 2-dimensional blob and
 (ii) a wiggly line, or something made up of a [finite!] number of wiggly lines.
 Then try to cover each of them with as few circles of radius 1cm as possible,
 and then with as few of radius 0.5cm as possible. (Don't worry about how exactly
 circular your circles are.) You should find that for (i) you need about four
 times as many half-size circles, but for (ii) you need about twice as many.)

  NB Don't make the line *too* wiggly. Roughly, to get the scaling coming out
     right you need the wiggles to be on a smaller scale than the size of the
     circles you use.

OK; what's happening here, translated into mathematician-speak, is that for
small enough values of r the number of circles (or balls, or boxes, or ...)
you need if they all have radius r, is about K/r^D where K is some constant
and D is the dimension of the object you're looking at.

Well, there are sets for which a relation like that holds WITH FRACTIONAL D,
and in these cases it seems sense to call D the "dimension" of the set.

 Example: That famous "snowflake" thing you get as follows: The first "stage"
 is just an equilateral triangle. The second is obtained by gluing on a 1/3-size
 equilateral triangle onto the outside of each side of the triangle; this looks
 like a Star of David. The third is obtained by gluing on a 1/9-size triangle
 to the outside of each line segment making up the new figure. And so on, adding
 more and smaller triangles every time.
 If you think of this as a "filled-in" shape, it has dimension 2. (This is easy
 to prove.) But what about the boundary? That actually has dimension 4/3, and
 this is sort of because if you look at it at 3 times the "magnification" you
 see 4 times as many line segments (at each stage you replace one line by four
 lines of 1/3 the length). If you like, try the same experiment as I suggested
 earlier!

This is the usual notion of fractional dimension; it is basically due to
Hausdorff. If you are reasonably expert with mathematics you could have a
look at "The Geometry of Fractal Sets" by Falconer, published CUP quite
recently (198x); if not, you'd perhaps do better to read one of the many
books on fractals -- certainly Mandelbrot's "The Fractal Geometry of Nature"
has quite a lot of discussion of dimension; there are more recent books,
some of which might also have the information you want.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
Newsgroups: sci.math