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

marcus

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 possess 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 doesnt 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/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)

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, lets say that the side length of the squares is [itex]s[/itex], and let's say that the number of squares that contain the shape is going to be [itex]N(s)[/itex].

Let's start by looking at a (solid) unit square. Clearly [itex]N(s) \approx \left(\frac{1}{s}\right)^2[/itex] for thesince the squares will cover an area of [itex]1[/itex] to [itex](1+s)^2[/itex].

Now, if we look at a unit line segment, we have [itex]N(s) \approx \left(\frac{1}{s}\right)^1[/itex].

If we were dealing with a unit cube and 'graph space' it would take up [itex]N(s) \approx \left(\frac{1}{s}\right)^3[/itex].

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
[itex]N(1)=1[/itex]
[itex]N\left(\frac{1}{3})=8[/itex]
[itex]N\left(\frac{1}{9})=64[/itex]
and, in general we know that
[itex]N\left(\frac{1}{3^n})=8^{n}=\left(3^{\log_3{8}}\right)^{n}=3^{n\log_3{8 }}[/itex]
so it's strongly suggested that
[tex]N(s) \approx \left(\frac{1}{s}\right)^{\log_3{8}}[/tex]

So the dimension of Sierpinsky's square should be [itex]\log_3{8}[/itex] 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

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:
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.

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

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

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.

Yes, but this is already well known. The question here is what if spacetime were a fractal.

marcus

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

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.