What does like a fractal mean, talking smallscale spacetime

[marcus]

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

A poster asks:

"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.

 

[marcus]

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.

 

[marcus]

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?

 

[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.

 

[Berislav]

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.

 

[marcus]

(edit:) Thanks for chipping in Berislav. I also did some digging in Wiki myself just now!

Wiki: "A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possesses infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or "broken". Before Mandelbrot coined his term, the common name for such structures (the Koch snowflake, for example) was monster curve."

What I get from this is that a fractal is NOT something with a repeating pattern, that repeats at all scales. According to the Wiki, A FRACTAL DOES NOT HAVE TO HAVE ANY PATTERN AT ALL. the repeating pattern stuff is only something about "the best examples"!

the repeating pattern jazz is merely a convenient strategem for constructing complexly kinky sets. It also has visually appealing results. But it also has mental economy: it saves a lot of effort. Imagine if you had to make up a different pattern at every step of the way down! So the POPULAR fractals are the ones that are also EASY to describe, where you use the same pattern repeatedly.

OK, so "fractal like" does not mean "HAS A PATTERN".

I remember attending a lecture by Mandelbrot in the late sixties, and IIRC he was just "this guy from Bell Labs" or something, and what he was mainly talking in what i remember was constructing stuff with fractional dimension----and also he was talking models of "noisy signal", and
"self-similar noise"

(this is why they pay you at Bell Lab in the sixties and seventies, because they think mathematical models of noisy signals will help make money and/or improve the lot of suffering mankind)

somebody who knows something about fractals really should take over. I may be quite wrong.

I think the question that had taken root in Mandelbrot's head at that era was "how do you make something that is really broken up, and really really branchy, and so screwed up it does not even have wholenumber dimension?"

but maybe asking that goes back much farther

Wiki again:
"Objects that are now called fractals were discovered and explored long before the word was coined. In 1872 Karl Weierstrass found an example of a function with the non-intuitive property that it is everywhere continuous but nowhere differentiable - the graph of this function would now be called a fractal. In 1904 Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. The idea of self-similar curves was taken further by Paul Pierre Lévy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve.

Georg Cantor gave examples of subsets of the real line with unusual properties - these Cantor sets are also now recognised as fractals. Iterated functions in the complex plane had been investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou, and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of the objects that they had discovered.

Aspects of set description

In an attempt to understand objects such as Cantor sets, mathematicians such as Constantin Carathéodory and Felix Hausdorff generalised the intuitive concept of dimension to include non-integer values. This was part of the general movement in the first part of the twentieth century to create a descriptive set theory; that is, a continuation of the direction of Cantor's research that was able in some way to classify sets of points in Euclidean space. The definition of Hausdorff dimension is geometric in nature, although it is based technically on tools from mathematical analysis...

Mandelbrot's contributions

In the 1960s Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. This built on earlier work by Lewis Fry Richardson. Taking a highly visual approach, Mandelbrot recognised connections between these previously unrelated strands of mathematics. In 1975 Mandelbrot coined the word fractal to describe self-similar objects which had no clear dimension. He derived the word fractal from the Latin fractus, meaning broken or irregular, and not from the word fractional, as is commonly believed. However, fractional itself is derived ultimately from fractus as well."

So I was wrong about his invention of the term. He called it fractal because it was FRACTURED, not because of fractional dimension. but a lot of people, like me, got the idea that he called it that because of fractional dimension.

I guess you can't get a set to have fractional dimension without taking a hammer to it and crushing and bending and kinking and beating the bejeezus out of it, if not actually breaking it up altogether. So the two ideas tend to merge.

I guess the thing I like so much about all this is the 1872 thing where the great Karl Weierstrass, the Victorian grandfather of so much 20th century mathematics, like functional analysis, where he observed this function that was continuuous (not actually broken) but so kinky that it was NOWHERE DIFFERENTIABLE. radically unsmooth at every point. Ach du Lieber, so wass ein verrücktes Funktion! I imagine him one morning still in his slippers and bathrobe, but already wearing a clean starched shirt, slightly near-sighted, with little spectacles.

This, I think, is the overall essential core idea: to be continuuous, and yet totally undifferentiable----nowhere having a welldefined slope----nowhere being fittable with a LINEAR APPROXIMATION----nowhere having a tangent line or plane or tangent surface of any sort.

a fractal person is someone whose clothes never look like they fit right. not even the most skillful tailor can cut and stitch together a smooth approximation to clothe him-----whatever you try, not only will there always be somewhere it doesn't fit him, things are worse than that, IT WILL NOT FIT HIM ANYWHERE.

it seems right that this should be a German idea, and a Victorian idea as well. Gilbert and Sullivan again. Lewis Carroll. When society is strict and proper then the mind is drawn to the maximally outrageous and eccentrics roam free in the forest.

But then Feynman!

Because when Feynman discovered the PATH INTEGRAL version of quantum mechanics where you describe the path of a particle getting from A to B by adding or superposing all piecewise linear paths, it turned out that the typical path was vintage 1872 Karl Weierstrass----THE TYPICAL FEYNMANN PATH IS CONTINUOUS AND NOWHERE DIFFERENTIABLE. oh wow. just like Weierstrass

and a spacetime is a path from one condition of space to another----from geometry A to geometry B. So now it seems right that if you look microscopically a spacetime should NOWHERE AND NEVER HAVE A SMOOTH OR FLAT APPROXIMATION
but if you look at large scale then you sort of gloss over these rebellious eccentricities and it looks like our familiar old smooth 4D spacetime, with everybody wearing a hat, coat, tie, tea at 4 and dinner at 9. The point is that you have to overlook the behavior at microscopic scale.

BTW none of this is supposed to be accurate or reliable. I am telling some intuitions about the essence of fractal

I am wondering why Loll is often saying that her CDT spacetime is "highly nonclassical" or "fractal-like" at small scale, and what that intuitively means. And as Loll has shown the dimension goes down steadily, from around 4 to around 2, as you go smaller and smaller scale. And I am trying to sense intuitively why that is what we should have expected. it is not surprising but somehow I suspect TYPICAL of fractals. but not sure about this.

I want a picture of Karl Weierstrass. I think he is the key player here.

 

[marcus]

Ah, Berislav is here! Take over Berislav.

here is Weierstrass the socalled "father of modern analysis"
http://www.stetson.edu/~efriedma/periodictable/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)

 

[NateTG]

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, let's 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.

 

[NateTG]

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.

 

[Berislav]

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:
http://www.math.lsa.umich.edu/mmss/coursesONLINE/chaos/chaos7/7.gif
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.

 

[neurocomp2003]

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.

 

[NateTG]

"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.

 

[Berislav]

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.

 

[marcus]

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)

 

[Spin_Network]

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

You bet!..I do?

 

[Hurkyl]

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:

A totally disconnected space is one for which the only connected, nonempty sets are single points. The rational numbers, for example, form a totally disconnected space, as does any nonstandard model of the reals.

However, a continuous, nowhere differentiable curve is connected. Why? Because it's the continuous image of [0, 1].

I suspect, though, that such a curve (at least if it's 1-1) is locally disconnected.

 

[Sam Owen]

um...yeah ok

it seems i don't know what a fractal is beyond the pretty 2d pictures you can easily source on the net.

So with regards to that and the split dimensions as CDT approaches ever smaller scales.

Is their model like slices of 3d or 2d planes lined up really close to each other like transparent onion skin books where as you add another layer the complexity of the model is increased depending on what information is on any particular page. Still 2d but with the appearance of depth ?

It just seems like they are saying that the smaller you look the more chaotic what you are looking at becomes until you really have no idea what it is unless you revert to 2dimensions cos if you didn't you would end up with an infinitely small 3d unit of spacetime which resembles 4d on a universally large scale...the appearance of fracticality

and it really doesn't matter what you look at, as at some stage microscopically you end up in the spaces between fundamental particles anyway

Once you add time to the mix time it becomes even more unpredictable as it seems no one has a handle on that concept either

Peter lynds from what i have read seems to think time doesn't exist just a succession of events so would that be like flipping the onion skin page at superluminal speed ?

I keep thinking about my 2 sets of 3d spaces interchanging in the same place but I don't quite know it's relevence yet so excuse the sidetrak.

 

[Spin_Network]

um...yeah ok

it seems i don't know what a fractal is beyond the pretty 2d pictures you can easily source on the net.

So with regards to that and the split dimensions as CDT approaches ever smaller scales.

Is their model like slices of 3d or 2d planes lined up really close to each other like transparent onion skin books where as you add another layer the complexity of the model is increased depending on what information is on any particular page. Still 2d but with the appearance of depth ?

It just seems like they are saying that the smaller you look the more chaotic what you are looking at becomes until you really have no idea what it is unless you revert to 2dimensions cos if you didn't you would end up with an infinitely small 3d unit of spacetime which resembles 4d on a universally large scale...the appearance of fracticality

and it really doesn't matter what you look at, as at some stage microscopically you end up in the spaces between fundamental particles anyway

Once you add time to the mix time it becomes even more unpredictable as it seems no one has a handle on that concept either

Peter lynds from what i have read seems to think time doesn't exist just a succession of events so would that be like flipping the onion skin page at superluminal speed ?

I keep thinking about my 2 sets of 3d spaces interchanging in the same place but I don't quite know it's relevence yet so excuse the sidetrak.

Take a look at my "fractal-inference" with respect to marcus thread here:https://www.physicsforums.com/showthread.php?p=689364#post689364


for begginers guide to the solution of CDT, I will be revealing something interesting, if I think anyone in CDT, or the recent papers marcus flagged (which I have not read yet), are getting the jist of things?

In the link above?..OR maybe I should stop teasing everyone and post the whole solution?

ACTUALLY..you know what, I am going to start the ball rolling,,the first picture is going online tonight!

 

[Hurkyl]

I suspect, though, that such a curve (at least if it's 1-1) is locally disconnected.

Actually, I'm wrong here. The function f that defines the curve is a continuous function on the compact set [0, 1], and thus its inverse is continuous... in other words, the fractal curve is homeomorphic to the unit interval. As such it's a manifold!

 

[mccrone]

"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?"

---

This discussions seems all back to front. The key property of fractals is self-similarity - over all scales of observation. So the "fractal" bit of the universe would be the 4D realm. The Plankian scale would be where the smoothness of fracticity breaks down.

A second problem here is that the fractal models being employed are background dependent. Even if imagined as spacetime making processes, they are processes "in a box".

What is the opposite of background dependent modelling? Is it background independent models? Or is it the more fruitful idea of "system dependent" modelling? That is a bootstrapping or semiotic approach where a system forms its own boundary conditions by constraint from within.

The difficulty for a background dependent approach to fracticity is that it offers no boundary cut-off. It must go to infinity in both directions (locally and globally). But a systems dependent approach does model the smallest and largest scales as event horizons. A cut-off does emerge as a feature of the growth of scale.

Anyway, the key point here is that the "fractal dimension" describes the "flat" middle ground of the system, not the break down at the boundaries.

To take the analogy of a coastline, it is fractal over the wide range of scales of observation where the appearance is of a ragged line. It looses this fracticity at the boundary cut-offs. At the most global scale (step back far enough from the earth) and all coastlines become smooth curves. Or zoom in far enough for the most local view and perhaps all we can see are smoothly curved grains of sand.

Fractal dimension describes an axis of scale symmetry that emerges between two scale limits. In a typical model, such as Brownian motion, the limits would be the zero-D particle and the 3-D box it is allowed to wander. The fractal dimension would then be a measure of the middle ground wanderings over "all scales" - all scales between these background specified limits.

 

[Chronos]

I encourage everyone to take a look at information theory to get a feel for fractals [and dimensionality concepts]. It really helped me. I'm interested in what you think about it, so I prefer not to prejudice your opinions. Here is a link:

http://www.faqs.org/faqs/fractal-faq/

 

[mccrone]

What's the connection with information theory here? Nothing really explicit in the faq.

Did you mean entropy and dissipative structure? Or even the infodynamic approach?

 

[Berislav]

A totally disconnected space is one for which the only connected, nonempty sets are single points. The rational numbers, for example, form a totally disconnected space, as does any nonstandard model of the reals.

There are also fractals with this property. The Cantor set, for example.
In any case you are right, it was my mistake to say that fractals in general are totally disconnected. The Koch curve for instance is obviously connected, but it has a dimensionality higher than one, which is what we do not want.

However, a continuous, nowhere differentiable curve is connected. Why? Because it's the continuous image of [0, 1].

Yes, you are very clever to note that. I stand corrected. But as marcus said:

And as Loll has shown the dimension goes down steadily, from around 4 to around 2, as you go smaller and smaller scale.

This means that in Loll's case the dimensionality decreases. Therefore the fractal must have a covering dimension of less than four. Thus it must be at least locally disconnected.

EDIT: It could be a discreate space (?)

 

[Sam Owen]

I don't understand how spatial or temporal dimensions in 4d reality go down steadily to 2d just by looking at things on a smaller scale !

is that 2 spatial dimensions or one and time ?

and what then of the matter occupying the space in 4d ?

how does one get 0.9 of a spatial dimension or 0.4 of one ?

Shouldn't there be a cut off point at which a dimension just drops off the radar at say Planck level ?

Is it like the idea of 2d slices reminiscent of those animated flipping pages where the image on each slice or page is slightly different but when flipped it gives the appearance of movement and depth ?

 

[marcus]

...
...
This means that in Loll's case the dimensionality decreases. Therefore the fractal must have a covering dimension of less than four. Thus it must be at least locally disconnected.

EDIT: It could be a discrete space (?)

sounds like we have a confusion here,
maybe a confusion of terminology

In my opinion, and as far as I know (not having sure knowledge), it does not have to be a discrete space.

It sounds to me as if you might be referring to a THEOREM WHICH I DON'T KNOW.

It could be that you are right and that the theorem applies here, and then I would like to know it. Or it could be that for some reason the theorem does not apply (perhaps because the definition of "dimension" does not agree, or because the conditions are not met), and in that case we should find out too.

So I would be glad if you would define "dimension" and "covering dimension" and "locally disconnected" and state a theorem.
Or it can be a conjecture of yours which might be a theorem.
You may have already done this in earlier posts in this thread!
In this case, to save the trouble of looking for them, please tell which posts.
==========================
I have to say that some post of Chronos (e.g. the FAQ) and Berislav and others teach me that I was wrong in some of my ideas of what "fractal" means.
At the start of discussion, I thought the "self-similarity" or repeating pattern was NOT ESSENTIAL to the idea but just a TECHNIQUE for producing examples, so that MANY EXAMPLES (but not all) would have this property.
But now I have been reading the correct definition.

I still think that "fractal-like" objects do not have to have this repeating pattern, but should only have the appearance of a true fractal, being very branchy or finely divided and seeming chaotic. I still think that this is more an intuitive, not precise mathematical, idea. But I may be wrong about this too.
============================

Now I will say what I think Berislav MAYBE COULD MEAN about "locally disconnected".

an imperfect but simple example is a long thin wire that one crumples into a ball or a wad.
this then inherits the ordinary 3D euclidean metric, and is a topological metric space with the metric topology

but if one takes a point in the space and a small (but not too small) neighborhood about this point, this neighborhood will be disconnected, typically very disconnected,

because sections of the wire will bend into it and then go out again, so the neighborhood of some radius will only contain a "U-shape" bit of that section of wire and another "U-shape" bit of some other section

to make the example better one should make the wire infinite long and infinite thin. but this just is for the idea of "locally disconnected"

SO SOMETHING CAN CONSIST OF A SINGLE CONNECTED COMPONENT AND STILL BE LOCALLY DISCONNECTED

(the mathematicians here all know this but I say it for everybody listening)

and fractals and TREE-LIKE stuff is even more this way. If you go out only a certain radius and take a photograph of what is just in that small neighborhood then you see something very very disconnected, but the
whole thing (not limited to the small neighborhood) can be all one piece.

======================

now when we were talking about DIMENSION in connection with Loll triangle CDT spacetime, there were two measures of dimension

hausdorff (you compare radius to volume)
spectral (you run an imaginary diffusion process, taking a random walk in the object and seeing how easily you get lost, and how hard it is to find way back home)

there may be several other possible definitions of dimension that we could use, I don't know them. Of course our sets are NOT VECTOR SPACE so the familiar vectorspace definition of dimension can not be used.

Berislav, if you state a theorem, please say what idea of dimension.

Also what is COVERING DIMENSION?

=====================

anyway this is how I am understanding what B. says at the moment, the Loll continuum is very nice connected. it is all one single piece of fabric.
but the fabric is wrinkled in such a way that it is LOCALLY disconnected.

==================
I said I would leave the discussion to others, in this thread, and I still want to do that, so as soon as I understand this locally disconnected business better I will get back out. have fun. thanks everybody, it is a nice discussion!

 

[marcus]

the wad of wire also illustrates how the (hausdorff) dimension can differ at different scales

at large scale the wad of wire will have an average density
so if you choose a point and consider spherical nbds around that point
if you DOUBLE THE RADIUS the weight of wire contained in the sphere goes up by factor EIGHT

this is the usual R³ behavior of volume and it means that the hausdorff dim of the space, at large scale, is THREE

but if you look at very small scale you see a simple segment of wire with empty space around it, so the hausdorff dimension is approx. ONE.
that is, if you double the size of the sphere, the weight of wire inside it only increases by factor of about two.------the volume is proportional to the radius

so at small scale the volume behaves as R¹ or some number close to that like R^0.9 or R^1.1 which will depend on if the wire is straight or slightly curved. But that is unimportant detail. at small scale in this example the volume only behaves EXACTLY like R^1.0 if the segment we are looking at is straight.

anyway in this example the smallscale hausdorff dimension is obviously APPROXIMATELY ONE.

so we have an example of a single connected topol metric space where the largescale dim is 3
and if you look at neighborhood of a single point with magnifying glass the dimension is 1

and it will be found that it varies between 1 and 3 as you enlarge the neighborhood, so at some radius it will be like 1.5 and at larger radius 2.1 and so on.

that is all I have time to say about the variable dimension business. ciao.

 

[Berislav]

So I would be glad if you would define "dimension" and "covering dimension" and "locally disconnected" and state a theorem.

By "covering dimension" I meant Hausdorff covering dimension. The meaning probably got lost in translation. Locally disconnected probably means disconnected but not totally.

Or it can be a conjecture of yours which might be a theorem.

It's just a simple idea of how I would handle physical laws on a disconnected spacetime:

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.
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.

 

[marcus]

hi B., I think I may be happy as long as we say that the spacetime can be
at the same time
both connected (one single connected component)
and locally disconnected

(btw it is some years since I was a student and I have a poor memory, so I can not be sure of what I say here. Maybe Hurkyl can help. must confess I feel too lazy right now to go look in a book.)

 

[Hurkyl]

This means that in Loll's case the dimensionality decreases. Therefore the fractal must have a covering dimension of less than four. Thus it must be at least locally disconnected.

I dunno... I can imagine a continuous fractal curve embedded in R⁵ that is comprised of a dimension 4 piece concatenated with a dimension 2 piece.


But I should point out I don't know anything about CDT, so I can't say anything about what it says.

 

[Chronos]

What's the connection with information theory here? Nothing really explicit in the faq.

Did you mean entropy and dissipative structure? Or even the infodynamic approach?

Agreed, the FAQ is a good place to start, but does not explicitly address information theory. Another link:

Information theory and generalized statistics
http://www.arxiv.org/abs/cond-mat/0301343

 

[mccrone]

Things may be even clearer if you continue on to physical models like Benard cells and Bak sandpiles. Or scale-free networks.

A fractal system that expands or dissipates at a steady rate has special edge of criticality features. I think this is where connections with spacetime models are waiting to be made.

 

[Berislav]

hi B., I think I may be happy as long as we say that the spacetime can be
at the same time
both connected (one single connected component)
and locally disconnected

I agree. I like your line and ball example.

I dunno... I can imagine a continuous fractal curve embedded in R5 that is comprised of a dimension 4 piece concatenated with a dimension 2 piece.


But I should point out I don't know anything about CDT, so I can't say anything about what it says.

I think that that might complicate things. Although, I too I am not sure whether CDT is about embedding in 5D space.

Maybe marcus could clarify.

 

[marcus]

...I think that that might complicate things. Although, I too I am not sure whether CDT is about embedding in 5D space.
...

I agree. I think CDT is not about embedding in any higher-dimensional space. At least when I have read CDT papers I never saw any reference to that.

I sometimes try to clarify and guess when I don't know, but both you and Hurkyl realize that I am just a watcher of CDT, and very far from being able to speak with certainty. I don't do research in it, I just read papers with limited understanding.

At this point, I would encourage anyone to ask anything about CDT (if they happen to be interested) even if it sounds naive or confused, and the others (myself included) can try to answer even if we make fools of ourselves. Or we can say we don't know. I don't think we can do any harm to ourselves or anyone else by trying to understand and talk about it.

the simplexes in CDT are not "equilateral" but one can imagine them in a 2D case as equilateral triangles, and I think it helps to think sometimes of putting some number N of equilateral triangles around a point, where N is not equal to 6. If N is less than 6, then one can imagine it embedded in familiar space (at least if N is more than 2) and there is positive curvature at the point.

but what if N is more than 6? this is a place where both embedding and imagination seems to have trouble. it may be here that one can best see how the possibility arises of very kinky contorted geometry

 

[mccrone]

"but what if N is more than 6? this is a place where both embedding and imagination seems to have trouble. it may be here that one can best see how the possibility arises of very kinky contorted geometry"

This kind of hyperbolic and hyperspheric geometry due to increases/decreases in edges is illustrated in Wolfram's big book. It was about the only part that really enthused me.

 

[Alamino]

I´m far from understanding too much CDT, but after reading some papers it seems to me that there is no higher dimensional embeding in CDT. It seems that the simplices are the spacetime and there is nothing but them (I mean, there is no more dimensions, no more space).

I´m not sure if what I´m going to write is too obvious... if it is, I apologize. But after reading the AJL paper

http://arxiv.org/abs/hep-th/0505113

what I could grasp from this dimensional reduction is that the picture of dimension is that dimension is something defined by the way (hypothetical) particles move inside spacetime. The spectral dimension measured comes from the probability of return to a point so what I can imagine is that for short distances (what in the paper is the same as for short times) spacetime is connected in some way that in average you can move only in two directions (2D) from some point, but if you do that for a long time (or long distances) the average behaviour is as if you could move in more. I think that it has something to do with the connectivity properties of the triangulation model, but I need to study much more about this to continue. I don´t feel so qualified at this time...