Yes, according to Professor Tim Palmer, University of Oxford, in this article HERE

As a system loses information, the number of states you need to describe it diminishes. Wait long enough and you will find that the system reaches a point where no more states can be lost. In mathematical terms, this special subset of states is known as an invariant set.

Because black holes destroy information, Palmer suggests that the universe has an invariant set. Complex systems are affected by chaos, and the invariant set of a chaotic system is a fractal. Fractal invariant sets have unusual geometric properties.

The invariant set postulate concerns the possible relationship between fractal geometry and quantum mechanics and in particular the hypothesis that the former can assist in resolving some of the challenges posed by the latter. It is underpinned by nonlinear dynamical systems theory and black hole thermodynamics.

If the Universe is a complex system affected by chaos then its invariant set (a fixed state of rest) is likely to be a fractal.

To sum up, nothing in the real world is a true fractal due to the impossibility of infinite self-similarity within a finite universe (unless you believe Nassim haramein). However, it is still useful to use the word “fractal” to describe objects that contain self-similarity, if only finitely.

I am still struggling to reconcile your word “superficial”.

I see the term “fractal” as equal to the term “circular” or “oval”. We use these terms without consideration if they are superficial. Can you reduce a circle into infinity, how about an oval, or a rectangle. At the Planck level are there circles or ovals?
These terms are descriptive of a generally recognizable geometric shape or function, not size. I see “fractals” on the same terms as any generally identifiable geometric shape or mathematical function.

If I attach a circle to a circle I have a fractal. An atom may well contain (subatomic) fractal properties and structures that are recognizable at a macro level.

I see fractals as fundamental mathematical constructs, not as “superficial” descriptions of objects, which fail at a certain level.

I am still struggling to reconcile your word “superficial”.

I see the term “fractal” as equal to the term “circular” or “oval”. We use these terms without consideration if they are superficial. Can you reduce a circle into infinity, how about an oval, or a rectangle. At the Planck level are there circles or ovals?
These terms are descriptive of a generally recognizable geometric shape or function, not size. I see “fractals” on the same terms as any generally identifiable geometric shape or mathematical function.

If I attach a circle to a circle I have a fractal. An atom may well contain (subatomic) fractal properties and structures that are recognizable at a macro level.

I see fractals as fundamental mathematical constructs, not as “superficial” descriptions of objects, which fail at a certain level.

Sorry, this seems to me as so much disjointed babbling, I have no hope of pulling it apart and addressing all the problems without getting sucked further down. It’s metaphysical tachyons all over again; no reward in trying.

questions:
a) is water an expression of fractal geometry? It is made up from a geometric form of identical (iterated) H2O molecules.
b) are H2O molecules constructed fractally. They are identical, not self identical, but was their creation a fractal function?
c) are “elemental particles” created by fractal function?
d) is cell division a fractal function?

I am still struggling to reconcile your word “superficial”.

I see the term “fractal” as equal to the term “circular” or “oval”. We use these terms without consideration if they are superficial. Can you reduce a circle into infinity, how about an oval, or a rectangle. At the Planck level are there circles or ovals?
These terms are descriptive of a generally recognizable geometric shape or function, not size. I see “fractals” on the same terms as any generally identifiable geometric shape or mathematical function.

If I attach a circle to a circle I have a fractal. An atom may well contain (subatomic) fractal properties and structures that are recognizable at a macro level.

I see fractals as fundamental mathematical constructs, not as “superficial” descriptions of objects, which fail at a certain level.

Sorry, this seems to me as so much disjointed babbling, I have no hope of pulling it apart and addressing all the problems without getting sucked further down. It’s metaphysical tachyons all over again; no reward in trying.

Well… you are making me think, so its not a total waste…

To sum up, nothing in the real world is a true fractal due to the impossibility of infinite self-similarity within a finite universe (unless you believe Nassim haramein). However, it is still useful to use the word “fractal” to describe objects that contain self-similarity, if only finitely.

Precisely. Just as there are no perfect triangles, circles and other geometric objects in the real world, nevertheless it is valid and very useful to use their approximations wrt to real objects in the rational understanding of reality.

a)Is it correct to say that “causal dynamic triangulation” (CDT) is a geometric function and that a “fractal” is the result of this function?

b)Can a fractal function be based on a different fundamental geometric instruction than “triangulation”?

c) If so, can you have a fractal which is the result of multiple (different) fractal instructions?

Not quite so. CDT is the method used to assemble spacetime using triangulation (with causality and time as fundamental rules). Triangles are just convenient building blocks which can efficiently approximate curved surfaces in computer simulations.

The only physically relevant information comes from the collective behavior of the building blocks imagining that each one is shrunk down to zero size. In this limit, nothing depends on whether the blocks were triangular, cubic, pentagonal or any mixture thereof to start with.

From a sidebar in the article:

By the authorsâ€™ calculations, the spectral dimension of spacetime shades from four (on large scales) to two (on small scales), and spacetime breaks up from a smooth continuum into a gnarled fractal.

Quantum spacetime may be like snow, which is fractal on small scales ..... but smooth and fully three-dimensional on large ones.

To sum up, nothing in the real world is a true fractal due to the impossibility of infinite self-similarity within a finite universe (unless you believe Nassim haramein). However, it is still useful to use the word “fractal” to describe objects that contain self-similarity, if only finitely.

Precisely. Just as there are no perfect triangles, circles and other geometric objects in the real world, nevertheless it is valid and very useful to use their approximations wrt to real objects in the rational understanding of reality.

I agree. But I think PC is arguing that infinite self-similarity is a fundamental quality of fractals, therefore you cannot call anything a fractal if it is not infinitely self-similar. I’m not so much of a purist , mainly because it leaves us with a lack of a term for things that are “superficially” fractal.

With an infinite number of iterations, we can continuously zoom in on this fractal and always find self-similarity. However, we don’t need to resort to infinite iterations to see my criticism of superficiality.

But I think PC is arguing that infinite self-similarity is a fundamental quality of fractals, therefore you cannot call anything a fractal if it is not infinitely self-similar.

A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity.

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals.

Exact self-similarity is not found in reality. That would be perfect scale symmetry.