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Hi guys!

Another question popped up that makes me wonder if I understand fractal dimensions correctly at all..

Here https://classes.yale.edu/fractals/Panorama/Astronomy/Galaxies/Galaxies.html   it is said, that the fractal distribution of galaxies on a small scale (<50million lightyears) is d=1.23 and on a large scale d=2.

how can the distribution of something in 3d-space be smaller than 2??
I really don't get that. doesn't that leave out one whole dimension, the one of space and put all galaxies somewhere between a 1d line and a 2d plane?

can anyone help me figure that out?

cheers,
chillheimer

A point is zero dimensional, regardless of the dimensionality of the space in which that point is embedded. A straight line is one dimensional, again regardless of the surrounding space.

In that same sense, fractals can have any dimension less than or equal to the surrounding space. (Fractal) Dimensionality is inherent to the object, not (strictly) dependent on the surrounding space.

yes - but:
how can a thing that has a 3 dimensional volume itself, like a galaxy be distributed in a 3dimensional space and that distribution (that needs x,y and z coordinates to specify a location) NOT be 3dimensional?

this doesn't make sense to me.

it is not filling the full space itself, so it must have a dimension smaller than 3. that seems obvious.
hm.. and if you take a single slice of space, galaxie distribution won't cover the whole slice either... hmm.. is that why its also below 2d??
is that the point?
but then again, if you only took a 1d line out of that slice, only afew points of that line would be part of a galaxy. so following that path it would even be smaller than 1.
I don't get it.. :hmh:

Just split it up into dimensionality of spaces and dimensionality of shapes/sets. For example the dimensionality of a cloud of point in a 3d (curved) space. Or comparable the dimensionality of cantor dust in a 2 dimensional space.

Remember fractal dimension is not actually about physical space.

So say you are using box counting to determine dimension, for a 3d fractal you would count 3d cubes.

By comparing how many cubes are needed to cover the fractal and  the making the cube smaller seeing how many more are needed you can calculate the dimension.

So fractal dimension technically is applied to any kind of object in any dimension.

So I suppose fractal dimension is a misleading term,  it's more fractal ratio.  Or an indicator how fractal an object is.


The is the best explanation I could find:

http://fractalfoundation.org/OFC/OFC-10-3.html

So the fractal dimension of a menger cube is 2.72

Galaxies are definitely by intuition not as fractal as a menger cube. But about as fractal as a coastline.
So you can guess it must be > 1.2 and far less than 2.7

Without worrying about the fractal aspect, think of a very long piece of string in our regular 3D space.  It's embedded in 3D and has a volume, but it's topologically (essentially) one-dimensional.  That is, if you're riding along the string, whether it's stretched tight or curled up into a ball, it only takes one number to describe your location along the string.  That's how an object can have 3D properties (volume) but also have a lower dimensionality when looked at in another way.  In terms of galaxy distributions, having a fractal dimension < 1 basically means that they're really sparse, more like discrete points than a coherent structure.  A fractal dimension between 1 and 2 suggests that there is some structure to the distribution, more than that of a piece of string, but less than that of a crumpled piece of paper.  As was said before, the fractal dimension is about the object, not the space in which it exists.

lets bring in tglads excellent table for defining fractal dimensions:
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=3841
(http://nocache-nocookies.digitalgott.com/gallery/3/853_16_10_10_2_29_36.jpeg)

that:
combined with this:
and it finally clicked.

thank you guys!  :D

the closer you are to an integer-dimension thess rough it is ...

Is that true no matter from which "side" you come?
I thought its more like 1.1 is not very rough but 1.99 is very rough

While fractal dimension does indicate how fractal an object is, it's also, in a sense, an extension of the standard concept of dimensions.

This video gives a pretty good explanation:
https://www.youtube.com/watch?v=g-csmdpq39A

I'll also try my hand at explaining it. Here goes...

Take a straight line (a 1D object) embedded in a 2D space. If that line has length x , it can be covered completely by taking x circles of diameter 1, and placing them end-to-end along the length of the line. Now if you multiply the line's length by k, it will take k times as many spheres to cover it (specifically, x*k spheres).

Now try embedding the 1D line in 3D space. Similarly, a line of length x can be enclosed by x spheres of diameter 1 (such that each point on the line is either inside a sphere or on the edge of a sphere). Multiply the line's length by k, and it takes k times as many spheres to cover it. Same rule as in dimension 2.

In general, for any non-fractal 1D-object, such as a line, embedded in an n-dimensional space, if you multiply the object's length by k, it will take k times as many n-spheres to cover/enclose it.

Now let's embed a finite plane (a 2D object) in a 3-dimensional space. Let's say this plane can be completely enclosed by n spheres of a fixed radius. If you multiply the plane's size by k, it will take k squared spheres to enclose it, as the area of a plane has a quadratic relationship to it size.

In general, for any non-fractal 2d-object, such as a plane, embedded in an n-dimensional space, if you multiply the objects size by k, it will take k^2 times as many n-spheres to enclose it.

Now, if you consider fractal objects, the rules change a bit. Let's imagine a space-filling curve, such as the Hilbert curve, filling a finite plane. The curve itself is 1-dimensional, but when iterated to it's limit, it's essentially equivalent to a plane. The number of n-spheres (where n is the dimension of the space it's embedded in) needed to cover scales quadratically with the absolute size of the curve. SO a space-filling curve, although  1-dimensional object, has fractal dimension 2.

A similar thing can be said about all fractals. Most fractals are not space-filling like the Hilbert curve, yet due to their roughness, they "fill" more "space" than a non-fractal object of the same dimension. Take the Koch  Snowflake. It's dimension is 1.26186. That means that, if you scale it up by factor k, the number of n-spheres required to cover it will scale by k^1.26186. So it's somewhere inbetween a line and a plane.

This sphere covering approach is essentially the same thing as box-counting.

It's more like, the closer the fractal dimension is to the object's topological dimension, the less rough it is. For example, the Hilbert curve has a topological dimension of 1 (it's a curve), but its fractal dimension is 2 (it fills an area), so it's very rough.