Oh the trouble with fractals!

[Cyclops]

I have a problem. You see I love fractals and chaos, and am fascinated by numbers, but I'm useless when it comes to maths!!
I mean I get the basics of how fractals are created but man this stuff scares me!

Let X be a metric space. If S\subset X and d\in[0,\infty), the d-dimensional Hausdorff content of S is defined by

    C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.

In other words, C_H^d(S) is the infimum of the set of numbers \delta\ge 0 such that there is some (indexed) collection of balls \{B(x_i,r_i):i\in I\} with ri > 0 for each i\in I which satisfies \sum_{i\in I}r_i^d<\delta. (One can assume, with no loss of generality, that the index set I is the natural numbers \mathbb N.) Here, we use the standard convention that inf Ø =∞. The Hausdorff dimension of X is defined by

    \operatorname{dim}_{\operatorname{H}}(X):=\inf\{d\ge 0: C_H^d(X)=0\}.

Equivalently, \operatorname{dim}_{\operatorname{H}}(X) may be defined as the infimum of the set of d\in[0,\infty) such that the d-dimensional Hausdorff measure of X is zero. This is the same as the supremum of the set of d\in[0,\infty) such that the d-dimensional Hausdorff measure of X is infinite (except that when this latter set of numbers d is empty the Hausdorff dimension is zero).

 
I just can make no head or tail of it all!

[David Makin]

Me too - a couple of years ago I asked my brother to get me a maths book about fractals for Xmas, he bought me "Chaos, Dynamics and Fractals - an algorithmic approach to deterministic chaos" by J.L McCauley from the Cambridge Nonlinear Science Series.
He slightly over-rated my maths education, I have yet to bring my maths to a level at which I can make head or tail of it.

[Cyclops]

Me too - a couple of years ago I asked my brother to get me a maths book about fractals for Xmas, he bought me "Chaos, Dynamics and Fractals - an algorithmic approach to deterministic chaos" by J.L McCauley from the Cambridge Nonlinear Science Series.
He slightly over-rated my maths education, I have yet to bring my maths to a level at which I can make head or tail of it.

I thought it was just me being stupid but even on the wiki I'm lost! There's an article dealing with fractal dimensions and it says how "one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension"
OK, so picking that apart to try and understand dit in pieces I clicked on toplogical dimension and got more confused:
"In mathematics, the Lebesgue covering dimension or topological dimension of a topological space X is defined to be the minimum value of n, such that every open cover of X has an open refinement in which no point is included in more than n+1 elements. If no such minimal n exists, the space is said to be of infinite covering dimension."


AAAAAAHHH!!!!

[iteron]

Working with the full and technically precise definition of the Hausdorff dimension is not going to be easy on you.

The definition is not short and easy.

It's not always necessary to have to master that level to work with fractals.

Besides it'll take you awhile to get to that point, and then you'll probably have to spend some time pondering the definition before you compute an actual measure.

I've seen the term 'fractal dimension' defined in several ways.  I think the most common (sometimes called the similarity dimension) is the following;




where N is the number of units.
P is the size of the object.
p is the size of the unit.

Because it's actually a measure of how an object fills space I would think that 'capacity' is a better term for this measure (a term used by some mathematicians).  It leaves the word 'dimension' out of it, which only serves to confuse, we can just say it's a measure of the objects capacity.

I mean what really is something with a 1.26 dimension? does such a dimension even exist?

The topological dimension is the natural dimension of an object that we usually think of, 1 (lines and curves) 2 (planes or surfaces), 3 (space, solid objects).

one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension

That's just a way to say that the dimension of a fractal is a fraction.

They say it that way because some fractal curves have an integer dimension of 2.  Even if the dimension is not a fraction the Hausdorff dimension is greater than the topological dimension, which is 1 for all the curves, and those curves are still considered fractals.

Regards,
Marco

[Cyclops]

Working with the full and technically precise definition of the Hausdorff dimension is not going to be easy on you.

The definition is not short and easy.

It's not always necessary to have to master that level to work with fractals.

Besides it'll take you awhile to get to that point, and then you'll probably have to spend some time pondering the definition before you compute an actual measure.

I've seen the term 'fractal dimension' defined in several ways.  I think the most common (sometimes called the similarity dimension) is the following;


<Quoted Image Removed>

where N is the number of units.
P is the size of the object.
p is the size of the unit.

Because it's actually a measure of how an object fills space I would think that 'capacity' is a better term for this measure (a term used by some mathematicians).  It leaves the word 'dimension' out of it, which only serves to confuse, we can just say it's a measure of the objects capacity.

I mean what really is something with a 1.26 dimension? does such a dimension even exist?

The topological dimension is the natural dimension of an object that we usually think of, 1 (lines and curves) 2 (planes or surfaces), 3 (space, solid objects).

one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension

That's just a way to say that the dimension of a fractal is a fraction.

They say it that way because some fractals have an integer dimension of 2.  Even if the dimension is not a fraction the Hausdorff dimension is greater than the topological dimension, which is 1 for all the curves, and those curves are still considered fractals.

Regards,
Marco

See that stuff is just way over my head-never  did algebra at school and don't know what log means! Its hopeless! Makes me look stupid!

[bib]

I'm not an expert, but I think that in simple terms, the Hausdorff dimension being higher than the topological dimension simply means that a fractal "fills" more or less the plan, if we talk about a curve (like the "border" of the Mandelbrot set). It also work in the 3D space, or any n-dimensional space. A curve's topological dimension is 1, it makes sense. It can be stretched to a line without breaking it. The more the curve is long, detailed, and convoluted (like the border of the Mandelbrot set), the more it will "fill" the plan and approach dimension 2. I think that Mandelbrot first named these shapes "fractional" because their Hausdorff dimension was a non-integer fraction (e.g. 1.26). So fractals (in the plane) are objects "between" a curve and a plan.

[kram1032]

@ bibÄs explanation:

maybe with the exception of space-filling curves which also are considered fractal but have a dimension of 3 (or n for the n-space)

though, that's the idea I'd say...

[Cyclops]

See I'm hung up on this topology business. Is the topological dimension the physical shape and form of an object? For example to put it in context I might say that a brocolli has a high topological dimension because its outline is highly complex. That about right?

[David Makin]

See I'm hung up on this topology business. Is the topological dimension the physical shape and form of an object? For example to put it in context I might say that a brocolli has a high topological dimension because its outline is highly complex. That about right?

No - the broccoli's surface, like any area, has topological dimension 2 but brocolli is complex as you say so the fractal dimension of its surface (by whichever fractal dimension you choose) will be greater than 2.

For instance the outline of the complex Mandelbrot Set is topologically of dimension 1 because it's topologically equivalent to a circle but because of its complexity it has fractal dimension >1 (if I remember correctly it's apparently 2 in fact - magnify the outline enough anywhere and it covers area as the border "line" gets infinitely dense).

[Tglad]

If its any help, I found this list helpful: http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
It doesn't state their topological dimension but I expect it its the following:
anything made of dots = 0
anything that is a curving line in 2d or 3d etc = 1
anything that is a curving/folding/crumpled surface = 2

so
topological dimension is what its made out of (dots, lines, surfaces...)
Hausdorff/fractal dimension is how much space it takes up.

[jehovajah]

Some rigour has to be introduced in defining what one is thinking about. but unfortunately academic protocol means that citations and acknowledgements have to be given in the oddest places. Mathematics has become encrusted with these references and citations and jargon. Some of these concerns are engendered by the discipline of the time. Bertrand Russel for example felt that this kind of explicitly referenced presentation might avoid paradoxical statements. However this type of statement is constructed and expected to be re-constructed by the reader and expositor following the syntax laid out but not explicitly. IT IS LIKE BUILDING A FLATPACK WITHOUT THE DIAGRAMS.

When Hausdorf was playing with these types of sets in which the important feature was that some way of measuring distance could be found for any 3  points or members of the set( So straight away he was thinking geometrically! and so topologically) He experimented with numbers on a number line then probably looked to see if it would generalise to the plane and then geometrical space. Gradually his definition became a code which allowed him to reconstruct more or less precisely the elements and objects of his exploration. Other colleagues who were interested in the same topic  came to recognise his code and to construct from it the objects of his dicussion. This is a bad habit mathematicians have fallen into, not explaining what they are doing simply and clearly as they go along. When I read Hamiltons paper on the theory of couples i found in him a clear and warm expositor who wrote with a n inviting voice that made it a pleasure to read and follow abstracted mathematics.

 

[jehovajah]

Sometimes a concrete example helps. So anyone want to define the topology of a cheesecake? Then how many ways can we define its dimension as 3 and what assumptions do we have to make to do this?

The fun thing is one way to define a cover for the cheesecake is to eat it with a spoon!!

[JackOfTraDeZ]

I have many books on this and in fact have much science/math background, and I can concur that as you get into the advanced stuff you will very quickly feel like you are in quicksand. But the basic math behind simple fractals is actually quite easy. And you really don't need to know much about it if using current fractal generating software, of which there are many many now. Don't be discouraged - jump in!

For the basic mandelbrot, download this program:

http://www.fractal-animation.net/progz/benchmark.zip

It has the code written so you can see.

[iteron]

Part of this trouble is a current over emphasis on abstraction, set theory,  and convoluted definitions.

I try not too get bogged down in it.  I'm no expert at all but active experience in mathematics is still more important.  I see more of that on these forums for example.

N J Wildberger has written an interesting article about this trend;

http://web.maths.unsw.edu.au/~norman/views2.htm

For me the objects of maths (numbers, points, etc.) remain substantial things in themselves.

[jehovajah]

The objects of maths can be any thing or process. The objective of maths is to party!!

If you remember that you won't go far wrong and you will see where mathematics as pushed today is going wrong.

Today we can explain more exactly with picture, video and diagram and application and programming what it is we are thinking about. We do not need to rely on the literary approach of the previous centuries. I am hoping that the mathematical snobbery that accompanied the introduction of the calculator and computer proofs has gone with the wind that touted it.

"What the hell are you on about?" is a good mathematical question always, even to yourself.