# Are fractals related to string theory?

• xeon
In summary, fractals are related to conformal field theory and number theory. Fractal strings are fractals which have fractal dimension and are used in mathematics.
xeon
I have just seen some pretty pictures of flame fractals with reference to string theory. I was just wondering if or where they fit in.

thanks

Hm, I'm not very well-informed about string theory but I don't know of anything in mainstream string theory for which fractals are relevant.

However, poking around on google, it turns out there is a specialized subject called "fractal strings". As far as I can tell just from reading what Google turns up (I may not be understanding this correctly!), this is about what happens if you study strings which are "made of" fractals.

Usually when people talk about a "fractal", they mean something which is produced by an iterative process and is self-similar. As far as I can tell "fractal strings" are using the older and more restrictive definition of "fractal", that of something (like a manifold) with "fractional dimension"-- or rather, something for which the two different methods, Hausdorff dimension and topological dimension, of measuring the "dimension" of a manifold or figure turn out to give different values. For example you can imagine something which is floating on the outer edge of the Mandelbrot inkblot. Topologically this outer edge is one-dimensional, a line. But because the shape of the edge is so odd and fractal-y, there is a sense (as measured by the "hausdorff dimension") in which something moving on that edge has more than one dimension of freedom of movement.

Fractal strings seem to be about what happens when rather than studying strings which are properly one-dimensional, you study strings which have fractal dimension-- strings whose Hausdorff dimension is less than one but more than zero, fractional. You could construct a string like this for example by taking a normal one-dimensional string and cutting sections out of it, as is done to construct the Cantor set.

However looking around, it sounds like fractal strings are not really used in physics or physics theories. Their main application appears to be that they are useful as mathematical objects in studying number theory. It sounds like an interesting subject though!

...I don't know if this is what you were asking about or not :)

There are physically interesting fractals embedded in 2D which are related to conformal field theory, which in turn is relevant for string theory. I once observed that fractal dimensions of several interesting objects in 2D, e.g. percolation cluster (D = 91/48), percolation hull (D = 7/4), self-avoiding walks (D = 4/3) and red links (D = 3/4) fit into the magic formula

D = (100 - n^2)/48, n integer.

This follows immediately from Kac' formula for c = 0 with half-integer values, using D = 2 - 2h. I later learned that Hubert Saleur had found the same formula a few months before me (it was really quite obvious at the time), but it nevertheless paid for a four-year postdoc.

There is of course an glaring limitation with this formula: it only works for fractals embedded in 2D (like a string theory worldsheet), but the physically most interesting fractals live inside 3D. This observation got me starting on higher-dimensionalization of the algebraic structures of string theory.

## 1. What are fractals?

Fractals are geometric objects that exhibit self-similarity at different scales. This means that as you zoom in on a fractal, you will see similar patterns and structures repeated.

## 2. What is string theory?

String theory is a theoretical framework in physics that attempts to unify all the fundamental forces of nature by describing them in terms of tiny strings rather than point-like particles. It is still a work in progress and has not yet been fully proven.

## 3. How are fractals related to string theory?

There is currently no direct connection between fractals and string theory. However, some physicists have proposed using fractal structures to model the extra dimensions in string theory, which may aid in understanding the theory.

## 4. Are there any experiments or evidence linking fractals to string theory?

As string theory is still a theoretical framework, there is currently no experimental evidence linking it to fractals. However, there are ongoing studies and simulations exploring the potential connections between the two.

## 5. Can fractals help us understand string theory better?

Fractals may play a role in helping us visualize and conceptualize the complex concepts of string theory, but they are not essential in understanding the theory itself. Further research and experimentation are needed to fully understand the potential relationship between fractals and string theory.

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