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What does "like a fractal" mean, talking smallscale spacetime

"What is meant by spacetime is a fractal, fractal like or has a kinky fractally structure ?

I know what a fractal is and what they look like, so is it an appearance of fracticality or actually fractal and does the reductablity of the pattern ever stop at a cut off volume ?

also how does one get half a spatial dimension or half a temporal dimension surely a half is still a whole when talking of dimensions?"

How do you reply? Does anyone want to take a shot at this?

Can you think of ways in which a topological space with distance function d(x,y) could be LIKE a fractal but not actually BE a fractal?

this is basically a challenge to one's intuition isn't it? How usable is your intuitive grasp of what a fractal is like?

In basic nonperturbative spacetime dynamics papers Renate Loll and others are sometimes saying that the picture of spacetime emerging from their work (in CDT path integral, in Reuter renormalized quantum Gen Rel) is FRACTAL-LIKE AT SMALL SCALE. So that is the context we are talking.

"FRACTAL-LIKE" is NOT A TECHNICAL TERM so strictly mathematically speaking it does not mean a blooming thing. This is a purely intuitive notion: LIKE (in some appropriate sense) a fractal.

historically, sometimes vague intuitive notions come first and then some helpful math-minded person devises a rigorous precise technical meaning later, that you can define mathematically and that everybody agrees seems to correspond to the original vague idea. But I certainly am not at that stage and I didnt see Loll or Reuter give a definition yet. so AFAIK we are still at square one meaningwise.

Does anyone else want to take a shot at this person's question? What does LIKE A FRACTAL mean to you? I will probably take a shot at it, but am not guaranteed to be able to provide the best intuitive take.
 PhysOrg.com physics news on PhysOrg.com >> Kenneth Wilson, Nobel winner for physics, dies>> Two collider research teams find evidence of new particle Zc(3900)>> Scientists make first direct images of topological insulator's edge currents
 Recognitions: Gold Member Science Advisor First off doesn't there seem to be a contradiction between saying: "I know what a fractal is." and then asking "How does one get half a spatial dimension or half a temporal dimension surely a half is still a whole when talking of dimensions?" I am not an expert about fractals so someone who knows fractals please correct me if I am wrong but I always thought that THING ONE about fractals is that they can have FRACTional dimension. So if I am talking some topological space which is LIKE a fractal then I am not going to be surprised if someone says the dimensionality of the thing is, like, TWO AND A THIRD, right? So right off I sense that something is wrong. Am I wrong about fractals? Isn't one of the basic facts that they don't need to have whole-number dimension? I never studied the things, so somebody else please step in here if you can help out.
 Recognitions: Gold Member Science Advisor by the way, here is the Wikipedia main page http://en.wikipedia.org/wiki/Main_Page it is I suppose always a good place to touch base before asking or trying to answer any question (sometimes it might be wrong or distorted, but still a good base to touch) anybody know what Wiki says about fractal?

What does "like a fractal" mean, talking smallscale spacetime

The number of dimensions isn't the only problem. It seems to me that if we wish a space to be self-similiar to an infinite scale we must dispense with the traditional definition of sets. This, of course, would mess up the topology. Any open set would contain an discretely infinite number of empty subsets, otherwise it wouldn't be fractalic, it would be a normal space.
 Here's what wikipedia says about (dis)connected spaces: http://en.wikipedia.org/wiki/Connected_space What's probably most important physically is that we wouldn't be dealing with manifolds, anymore. The most important building block of spacetime would become a set, instead of a point.
 Recognitions: Gold Member Science Advisor Ah, Berislav is here! Take over Berislav. here is Weierstrass the socalled "father of modern analysis" http://www.stetson.edu/~efriedma/per...le/html/W.html I am surprised to see that Weierstrass didnt wear a beard! However notice the high starched collar which gets up under the ear-lobe. "Though Weierstrass showed promise in mathematics, his father wished him to study finance. So after graduating from the Gymnasium in 1834, he entered the University of Bonn with a course planned out for him which included the study of law, finance and economics. Weierstrass was torn between the subject he loved and the subject his father wanted for him, and he spent 4 years of intensive fencing and drinking." obviously the fencing and drinking is the important thing to know about Weierstrass, but additional detail can be found in Wiki http://en.wikipedia.org/wiki/Karl_Weierstrass BERISLAV HAS SHOWN UP now it is his turn, I'm out of here. edit: And NateTG too (I see from the next post). Thank's for chipping in! (best explanations come from several)
 Recognitions: Homework Help Science Advisor The mathematical notion of fractals is associated with a particular notion of dimension: Let's say we have a piece of graph paper, and we draw some shape on it, and then count the number of squares that contain any the shape. So, lets say that the side length of the squares is $s$, and let's say that the number of squares that contain the shape is going to be $N(s)$. Let's start by looking at a (solid) unit square. Clearly $N(s) \approx \left(\frac{1}{s}\right)^2$ for thesince the squares will cover an area of $1$ to $(1+s)^2$. Now, if we look at a unit line segment, we have $N(s) \approx \left(\frac{1}{s}\right)^1$. If we were dealing with a unit cube and 'graph space' it would take up $N(s) \approx \left(\frac{1}{s}\right)^3$. You should notice a correlation between the exponent, and the dimension of the object. Now, let's take a look at something a little more interesting, say a unit sierpinksy square: http://www.2dcurves.com/fractal/fractals.html Now, we have $N(1)=1$ $N\left(\frac{1}{3})=8$ $N\left(\frac{1}{9})=64$ and, in general we know that $N\left(\frac{1}{3^n})=8^{n}=\left(3^{\log_3{8}}\right)^{n}=3^{n\log_3{8 }}$ so it's strongly suggested that $$N(s) \approx \left(\frac{1}{s}\right)^{\log_3{8}}$$ So the dimension of Sierpinsky's square should be $\log_3{8}$ which concurs with the link above. Although Sierpinsky's carpet is clearly self-similar, it's pretty easy to see that if, instead of pulling out the middle sub-square at each step, you choose one of the sub-squares at random, you'll get a structure that's not self-similar, but still fractal.

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 Quote by Berislav The number of dimensions isn't the only problem. It seems to me that if we wish a space to be self-similiar to an infinite scale we must dispense with the traditional definition of sets. This, of course, would mess up the topology. Any open set would contain an discretely infinite number of empty subsets, otherwise it wouldn't be fractalic, it would be a normal space.
It's clear that, for example, the real, or rational, numbers are self-similar and there are absolutely no problems with topology for either of them. Even if we look at properly fractal sets - for example Cantor's set - there aren't any problems for set theory.
 I must say that I find this result by Loll to be very interesting. Fractals, AFAIK, are structures which are maximally self-similar. The importance of this is, as marcus mentions, that they are continious, but not differentiable anywhere. Spaces with this property are called totally disconnected. Here's another article about such spaces: http://mathworld.wolfram.com/DisconnectedSpace.html Fractals are typical examples of such spaces. Constructing physics on such a space would be difficult. I'm unfamiliar with Loll results (maybe someone can provide a link). Before I heard about this I was playing with the idea of physics on a discrete space. My idea would be to make physics set dependant. Instead of coordinates I would use sets for the gauge constraints. On some set we would have more physical degrees of freedom than on it's subsets. For instance, there is a classical particle on a Cantor set: Cantor set Taking the entire set to be the physical space the particle can move as if in normal one dimensional space. But if we take the singleton set to be physical that degree of freedom becomes gauge and the particle can't move anywhere.
 from a computational standpoint i was under the impression that fractals were ~automatas...you have a set of fundamental equiations that you iterate/recurse/process/mechanics/evolve/run over time to create complex patterns of some (LOD)level of detail. The rules could be based on pseudorandomness(plant modelling in 3D graphics) or concrete rules in which case there would be a high degree of self-similarity. but then again i've only generated some fractals. The pictures on wiki are kinda cool...the microwave burned cd. I find Gary Flake's book " Computational Beauty of Nature" has some well defined computational terms. I think the author of the article maybe chose the wrong computaional term to use.

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 Quote by marcus "FRACTAL-LIKE" is NOT A TECHNICAL TERM so strictly mathematically speaking it does not mean a blooming thing. This is a purely intuitive notion: LIKE (in some appropriate sense) a fractal. historically, sometimes vague intuitive notions come first and then some helpful math-minded person devises a rigorous precise technical meaning later, that you can define mathematically and that everybody agrees seems to correspond to the original vague idea. But I certainly am not at that stage and I didnt see Loll or Reuter give a definition yet. so AFAIK we are still at square one meaningwise. Does anyone else want to take a shot at this person's question? What does LIKE A FRACTAL mean to you? I will probably take a shot at it, but am not guaranteed to be able to provide the best intuitive take.
This is very much speculation.

Let's say that we have a system which is a stochastic process in from a state space to itself, and that, from any particular state, the system has a various probabilities (possibly zero) to transition to another state. With appropriate probabilities, the expected behavior of the system would be fractal noise.

 It's clear that, for example, the real, or rational, numbers are self-similar and there are absolutely no problems with topology for either of them. Even if we look at properly fractal sets - for example Cantor's set - there aren't any problems for set theory.
Yes, you're right. I meant that the topology wouldn't be trivial anymore and that would cause problems, not that the space wouldn't be topological anymore.

 Let's say that we have a system which is a stochastic process in from a state space to itself, and that, from any particular state, the system has a various probabilities (possibly zero) to transition to another state. With appropriate probabilities, the expected behavior of the system would be fractal noise.
Yes, but this is already well known. The question here is what if spacetime were a fractal.

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 Quote by Berislav Yes, but this is already well known. The question here is what if spacetime were a fractal.
hi Berislav
you asked for a reference describing the CDT spacetime
I always have the link to the main paper in my sig

your question "what if..." if you apply it to CDT is really
"what if spacetime looked like ordinary familiar 4D spacetime at our scale, and the scale of things like atoms and quarks and stuff, but what if it also, down at planck scale (which is much smaller) began to bend and wrinkle and fluctuate violently so that it resembled a fractal in some sense? what would this mean for physics?"

I admit that it would be very difficult to detect this! It is hard to imagine an experiment that could detect this "non-classical" or "fractal-like" geometric behavior at very very small scale. It does not seem to have any consequences for physics that one could see how to observe, at least naively.
But perhaps someday a clever person will think of some observation to test it (like some very gradual effect on light propagation over long distances, I don't know)
 How do you reply? Does anyone want to take a shot at this? You bet!..I do?

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