What is dimension?

how can you mathematically define dimension?

why saying we are living in a 3D world?

how is it like in a 2D world and a 4D world?

regarding Shadows theory (sorry I can not force myself through all the reading), shadows theory disregards the existence of time dimension (does it? I just vaguely understand it) by the existence of fourth dimension and curvature of this with the other 3.

if so, how can we have sense of order and sequence of actions?

How can you define motion in shadows theory? Cause motion is basically being at different place at different time.


Thank you.

A degree of freedom of motion. The number of data points you need to fully describe the motion of an object.

I've never seen a formal axiomatic definition.

Technically we live in a 4D world as time is a dimension. Think x,y,z,t or <u>i</u>,<u>j</u>,<u>k</u>, t.

A 2D world would be very plane. We already live in a 4D world, exciting isn't it.

Don't confuse the delusional ramblings of Jeff Ocal to anything resembling physical reality. Shadows is the random organisation of clever sounding words to make it look like it is scientifically valid. You could achieve the same result by chopping up some paper with big words on it and pulling them out of a scrabble bag. It might be more intelligible as well.

Please E-mail any questions directly to the author. Rather than cluttering up these forums with psychobabble and claptrap.

well, I was talking about spatial dimension. how do things look like in 4 spatial dimensional (5D with time) world?

I can not image how things look like in 1 spatial dimensional world? just points or rod-like structure basically?

I dont feel the definition of demension is concrete enough. kinda too qualitative

Why do you think it would necessarily feel any different?
Basically.
The definition of 'dimension' is a precise, rigorous mathematical concept. It is not qualitative at all.

- Warren

A zero dimensional world would be a point.
A one dimensional world is like a line.
A two dimensional world is like a plane.
A three dimensional world is like a space.
A four spatial-dimensional world is unlike anything we can visualise, although we can work out some of its properties.

and who is gong to introduce fractal dimensions?
oops I just did.

a space is n dimensional iff there is a local homeomorphism between that space and Rn. this is not the most general definition, but it is a good one. it is a topological invariant.

hausdorff dimension is the mathematical name for what you know as fractal dimenision.

Thanks for clearing that up for us, lethe.

no sweat, james. actually, my answer before was a little too concise. let me explain that in a little more detail...

the definition i gave above is the definition that i think is the most common and most useful, let me explain what it means.

a manifold is a space where you can find a local homeomorphism to Rn, for some n. Rn, is euclidean n dimensional space, which is defined as n direct products of the real line. a homeomorphism is a mapping that takes close points on the domain space to close points in the range space, and vice versus. also this mapping must be bijective, i.e. maps each point to exactly one point, and covers the whole range. a local homeomorphism means that for any point, i can find a homeomorphism of some neighborhood, some small interval, to Rn. it is not required to have one mapping that is good for the whole space, and so local homeomorphism is more lenient than regular homeomorphism.

some examples: the 2-sphere. given any point on the sphere, i can pick a small region around that point, and make a 1-1 onto map to the euclidean plane. therefore the sphere has dimension 2, by definition. but be aware, i cannot find a single mapping that will take the whole sphere to a plane, and still preserve the closeness of the points, in other words, it will not be a continuous mapping. try to think of a way, but i assure you it does not exist. the the sphere is not homeomorphic to the plane (as you might expect)

on the other hand, i can very easily find a homeomorphism from a paraboloid, or a cone, to a plane. they are homeomorphic spaces (also of dimension 2. i can easily let my global homeomorphism be a local homeomorphism as well)

if one neighborhood is homeomorphic to Rn, and some other neighborhood is homeomorphic to Rm, it is not too hard to show that n=m. thus dimension is unique.

if you are wondering how you axiomatically define a notion of "closeness" that i keep referring to, well you should study some topology. it is a really interesting branch of mathematics.

now, now. we all know he will work the math out, simply because its so elaquent it must work. We're just waiting for a little while.


I introduced then here http://www.sciforums.com/showthread....0&pagenumber=6

but no one seemed interested

maybe its time we start a thread on it. I'm somewhat confused on how the different methods of determining the fractal dimension.

so that is the definition of a manifold, and its dimension.

but i mentioned that it is not the most general definition available. you see, requiring my manifold to be homeomorphic to Rn is a bit more restrictive than i might require, for an abstract space. i might want to work with more abstract spaces, where a relationship to Rn is not easy, or possible, to find. or maybe i just don t want to have to work with something as specific as the reals. such abstract spaces can still have dimensionality. i will give the definition, but it is a bit harder to understand than the manifold definition.

A space X is finite dimensional if there is some integer n s.t. for every open covering A of the space, i can find a refinement B, of A, with order at most n+1. the smallest such integer is the dimension of X.

a covering of a space is a collection of open sets whose union is the whole space. a refinement of an open covering is a new covering every element of which is a subset of one of the elements of the old covering.

it can be shown without too much difficulty that an n-manifold has topological dimension of at most n. it is a bit harder to show that an n-manifold has topological dimension of exactly n, but it can be done.

as you can see, the manifold s dimensionality is more intuitive than the topological dimension. they are both topological invariants, and when they are both defined, they are equal.

you should be able to see how specifying n copies of the real lines is intuitively like specifying that you have n degrees of freedom. you can easily define an inner product on Rn such that these different degrees of freedom are perpendicular. so if you live on an n-manifold, at any time you have n different orthogonal directions you can move in.

to see how topological dimension leads to the notion of degrees of freedom is a bit harder, and requires some study. but next i will talk about Hausdorff dimension, and we will see the concept of coverings again. they are important to the notion of dimension.

now i have never seen a rigorous topological definition of hausdorff dimension, but here is how you calculate it.

your space is a subset of Rn. pick n-dimensional boxes that cover your space, and let them have edges of length e. then the hausdorff fractal dimension is log(N)/log(1/e) where N is the number of boxes, and i think you sometimes have to take the limit as e goes to zero. but not always.

i am not sure that this dimension is a topological invariant. but in the case that your space is a non fractal manifold, then this fractal dimension will give you the same number as the manifold dimension, and the topological dimension.

theres also at least 2 other methods for calculating the dimension. I think the method you described does not work for plama fractals

what is a plama fractal? i must admit, i m no expert in fractal geometry. i just know a bit of the basics.

why doesn t this method work for plama fractals?

edit : I finally tracked the site down I was looking for

I'm guessing the reason is because of the "random" factor. After reaing the plasma link, it seems the roughness factor would determine the dimension. Then the dimension would be set to that of clouds.

I'm a bit dissapointed in the site, it seemed more informative before. But it was set up in such a fashion that if you found yourself linked there in the middle, it took a while to get to a menu because there was only a forward / back button and it was about 50 pages. That should be a good start for you.

AHA!!!!! yes! I found it! I'm not going crazy. go here

http://library.thinkquest.org/26242/...orial/ch4.html

Plasma fractals are perhaps the most useful fractals there are. Unlike most other fractals, they have a random element in them, which gives them Brownian self-similarity.



methods:

1.self-similarity
The similarity method

eD = N

D = log N / log e

the Koch Snowflake. In it, you can see four identical snowflakes (N = 4). Each of them is 1/3 of the entire snowflake, so e = 3.
Calculating the fractal dimension, we get: D = log 4 / log 3 = 1.26

2. Geometric Method

WHAT’S WRONG WITH SIMILARITY METHOD?
The similarity method for calculating fractal dimension is great if you have a fractal composed of a certain number of identical versions of itself. However, try using it for the coast of Britain. That’s impossible because all lines there have different sizes and require different magnifications. And we wouldn’t suggest counting them either!


There is a simple way out of this. We know that a true fractal has an infinite amount of detail. This means that magnifying it adds additional detail, which increases the overall size. In non-fractals, however, the size always stays the same. If you graph log(size) against log(magnification) you get :fractal dimension = slope + 1

The geometric method can be used very efficiently for natural irregular shapes that exhibit Brownian self-similarity. It was used to calculate dimensions of coasts, borderlines and clouds.




3.Box-counting Method

Both, the similarity method and the geometric method of calculating fractal dimension require you to measure the size. Box-counting method is very useful for natural shapes that are hard to measure, especially bacteria cultures





This is a quick summary of the important notes, and when to use what method.


taken from "fractals unleashed" http://library.thinkquest.org/26242/...orial/ch4.html

homepage http://library.thinkquest.org/26242/...=1&tqtime=1206

happy chaos!

Clouds: plasma fractals

Usually, to prove that something is a fractal it is enough to find its fractal dimension. For something like a cloud it is the best to do it using the geometric method. Obviously, it is not done by measuring the actual cloud, but by measuring its 2D projection, which is the shade.

We can make several measurements of the cloud’s perimeter using different magnifications. This is achieved by using different sized "yardsticks." If, let’s say, our yardstick is 1 kilometer long, the magnification is higher and measurement would be more exact than the one where the yardstick is 10 kilometers long. We also know that in fractals, more detail adds additional irregularities, which adds to the measurement. If we graph log(magnification) against log(perimeter) we should get a line with a positive slope since the perimeter of fractals increases with magnification.

By adding 1 to the slope (see geometric method) we find the fractal dimension. According to the findings of Lovejoy in 1981, the fractal dimension for most clouds is about 1.164.

Now, having proved that clouds are fractals, it would be good to try using fractals to generate computer models of them. We know for sure that, since clouds are very irregular, we have to use fractals that are random and have Brownian self-similarity. The best ones to use are plasma fractals

We can control how fragmented the clouds are by changing a parameter in plasma fractals called roughness


http://library.thinkquest.org/26242/full/ap/ap3.html