Possibly, but I don’t know if we know what or why. I should warn you that I have a strange take on this but it is one that is informed by my background in physics and one that, I think, is meaningful.

Let’s start with what a fractal is. A fractal is, basically, a self-similar structure. That means that if you zoom in on it by some amount, it looks the same as the original unzoomed version.

As mathematical objects, these things are pretty cool. But fractals are self-similar when you zoom in on a fixed scale, say, by a factor of 3. One could also imagine a self-similarity when you zoom in on any scale. I am going to re-interpret your question a bit to think about this extreme self-similarity — the self-similarity of things that are equal on any scale, not just a fixed set of rescaling. In doing this, I am losing some of the aspects of fractals (infinite length containing a finite area, etc.) which also seem deep and meaningful, in order to capture one aspect: self-similarity.

This kind of self-similarity is called dilational symmetry or scale symmetry or some such, and it happens in the universe everywhere, for reasons that are murkier than most people suspect. Here are some simple examples:

1. Phase transitions: Some transitions between phases (like when water boils) are “first order.” Bubbles of gas having a definite size nucleate and grow. Other phase transitions are “second order,” and these have very special properties. There is a specific temperature and pressure at which a liquid and gas appear indistinguishable. At such a point, called a critical point, there are large fluctuations of density. The way we understand the physics is by using the fact that the liquid/gas becomes scale-free — it looks the same (statistically) when you zoom in by any amount. It turns out that this fact gives us powerful tools for predicting things about phase transitions. The fact that these critical points exist and are scale-free is a lucky break for physics. Our current understanding of the standard model includes at least one of these second order phase transitions.
2. Singularities: Singularities are scale free. Singularities that occur in nature, like the point at which a stream of water breaks up into a droplet, take shapes that are locally scale-free.

1. Dimensional analysis: One of the most powerful tools for understanding anything is dimensional analysis. However, if you think about it, you will realize that dimensional analysis is actually a statement about self-similarity in the universe. Why? Because if you have an equation where the left-hand side has units of length, then the right-hand side also has to have units of length. That way, if you decided to measure length in “quora” instead of “meters” and there are two “quora” per meter, then you can multiply both sides of the equation by 1/2 and everything would still work. That is a kind of scale-symmetry too, and you can use it to prove some weird and powerful things. For example, do you know Kepler’s 3rd law?

The orbital period of a planet can be related to the length of the major axis of its orbit. What a crazy, crazy law. Let’s start by thinking about what the orbital period should depend on. Surely the orbital period gets longer if the orbit is bigger (there is more distance to travel, right?). It also has to change with the strength of gravity, [math]G[/math], and the mass of the sun, [math]m_{sun}[/math]. And there really isn’t anything else. Now suppose you need to arrange these things together into a dimensionless combination. With some effort, you can convince yourself that [math]G m_{sun} \propto M^3/T^2[/math] because that is the only combination that works out and matches units. In other words, if you double the size of the orbit, you have to multiply the period by [math]\sqrt{2^3}[/math]. These kind of predictions are all over physics. This guy,

(Pierre-Gilles de Gennes) was famous for using them to understand very complicated systems like polymers. These are systems that are really just profoundly complicated. Too complicated to perform direct computations on. And yet, if we start thinking about the scale symmetries that exist in those systems, we can suddenly make a lot of quantitative progress.

Do fractals tell us something deep about reality? Almost assuredly they do. Do we know exactly what that is? No. What we know is that if we start from a theory that is valid at very small scales and we start trying to derive a theory from it that is valid at larger scales (a process called coarse-graining) we end up, automatically, with theories that exhibit the self-similarities that I described. I think this is a deep fact about the universe. It means that we don’t need to understand everything in order to understand something. In other words, we don’t need to know string theory to make a toaster because we can derive theories for toasters by taking advantage of the self-similarity of physics.

Sometimes I feel like Tom Hanks in Forrest Gump. By various odd circumstances I have met many world renowned mathematicians and scientists, a group that I in no way shape or form remotely resemble.
Benoit Mandelbrot is one such acquaintance, I met him at MIT when he gave a presentation, on fractals. (He was discussing fractional Brownian motion at the time. He sinced passed away in 2010.)

With this weakest of all possible credentials I will tentatively propose an answer. Hopefully we soon will have a string theoretic Quoran to outdo me here.

If I understand Beniot's book correctly, Fractal Geometry of Nature rev edition, Benoit B. Mandelbrot (9780716711865), fractal theory is about shapes in ordinary dimensions. Much of the discussion these days in physics is about actual physical geometric dimensions, see Lisa Randall Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions: Lisa Randall: 9780060531096: Amazon.com: Books.

There is a big difference. If I say I have 10 credit cards in my wallet I need a 10 dimensonal vector to describe the credit card numbers. If my wallet lies in a 10 dimensional space that our 3D world is embedded in, then my wallet might appear and disappear as far as my 3D world is concerned. It might pop up in YOUR pocket, if your 3D coordinates intersect a different place in the 10D world.

Fractals, loosely speaking, is about the credit card analogy.

String theory is more like the latter.

Cheers.

It’s quite simple. The universe appears as a fractal because the mechanics of the methodologies by which we perceive the universe are themselves arranged as fractals.

In short, the human body (and especially the brain) exhibits self-similarity across architectural magnitudes.

This is most easily demonstrable through a consideration of embryology. Our anatomy arises from the complexity inherent in cellular replication. Really, that’s all that needs to be said. A single-celled (hybrid) Zygote contains all of the necessary functionality for the development of an independent human organism. Hence, the apparatuses contained within that most basic form constitute the basic informatic conditions for ‘a human’. Indeed, peripheral contributors such as the uterine environment and epigenetic influences are important, but it is most evident that a zygote is (in fact) the be-all-and-end-all of human procreation. Hence, it is a fractal function manifest in biological terms.

In neurological terms the patterns are even prettier. See Fractals in the Nervous System: Conceptual Implications for Theoretical Neuroscience. Here’s my favourite example, a section of the arbor vitae that characterises the cerebellum (which seems to be gaining a new function every month - Learning to expect the unexpected: rapid updating in primate cerebellum during voluntary self-motion

So, if the world appears as a fractal it is most likely because we are made of fractals…but doesn’t that mean that ‘the world that made us’ is inherently arranged that way such that out anatomical assembly from gluons to glia merely follow that modus operandi? Well, not evidently - because fractal similarities only become describable at certain degrees of simplicity. Do we believe in a quantum world? If so, then what I like to call the ‘pixelisation point’ (at which macroscopic relationships and the geometries they describe become irrelevant) becomes the boundary that no universal fractal can cross. A fractal function cannot be both probabilistic and deterministic simultaneously. A sierpinski square carpet cannot also be a triangular one a la Schrodinger’s cat. Fractals are all about consistency and quantum cats piss on that idea like their larger counterparts do on macroscopic tapestries. However these virtual villains and their Newtonian neighbours are only comparable as far as that terrible joke allows.

In summary, when describing ‘the universe’ the answer ‘it is’ is always inextricable from ‘we are’. So it is with fractals. Just like the natural numbers, if they exist objectively such that our models of reality emerge necessarily from the functions that construct us then the universe becomes self-informatic and all our operations meld into an ultimate constructive existence that constructs representative manifestations of itself across orders of magnitude and time. This deterministic deism may also appear mildly attractive but to end thieves rant and leave some food for thought I will simply state the Nonsense Codon that halts the proliferation of any such pantheistic philosophy.

In a universe that begets itself by computational complexity, irony cannot possibly be perceived since no doubt as to semantic significance can be accomodated.

Because fractals can used to explain a lot of phe phenomena of our daily life.

Like the following using fractals to generate mountain like surface:

Or generate tree with fractals:

In fact, self similiar property is widely seen in our daily life. Like leaves,

lighting:

sea shell:

snow flake:

and many many more.

So it is the most common relations in our daily life. We need to have proper tools to understand it. That's why fractal is so important.

One of main areas in Mathematics where fractals are heavily used is on the study of dynamical systems, which is actually unsurprising since many fractals are actually defined using some kind of dynamical system. I could use the Mandelbrot (the set of parameters [math]c[/math], such that the orbit of [math]0[/math] for [math]f(z) = z^2+c[/math] is bounded) and Julia sets as one example, but I’ll take you to a different journey that will start in Sweden, go through France (a couple of times), Brazil, and actually the whole Solar System. Buckle up and we’ll soon be gone!

Our story starts in 1887, when king Oskar II of Sweden decided to sponsor a prize, in honor of his 60th birthday to whoever could prove whether the Solar System was stable.

Since Newton, a lot was know about a system of two masses in space, on which the only force acting is gravity. Their orbits are either elliptical with their center of mass at one the focii, or their initial relative velocity is so high that they escape from each other’s gravity. If you add a third mass, however, the system goes from very simple to very complicated. This is know as the n-body problem, and some of the world’s brightest minds had tried to crack it for centuries, but it stubbornly resisted unscathed.

Enter our first hero, Monsieur Henri Poincaré.

If there was someone that could claim a shot at that prize, it was Poincaré. 32 years old at the time, he was widely recognized as one of the brightest young mathematicians in France, and would later establish himself of not only one of the greatest genius of his generation, but one of the best mathematicians ever.

And, make no mistake, he did win the prize. What he didn’t do, though, was to prove that the Solar System was stable, or that it was unstable, for that matter. What he did do was to explain some of the key reasons why the n-body is so hard, and create the tools that eventually allowed future mathematicians to have a much deeper understanding about this and many other problems. In the process, he single-handedly created the whole field of Dynamical Systems.

To understand what Poincaré did, and how the Fractals will enter our story, we need to take a step back, and understand what a dynamical system is. Essentially, dynamical systems are systems that “evolve over time”. The aforementioned n-body system is an example, as are a pendulum, the weather, the flowing of a river…

We usually study two kinds of dynamical systems, depending whether we consider the time to be continuous ([math]t \in \R[/math]) or discrete ([math]t \in \Z[/math]). The examples I just mentioned are all continuous, for discrete dynamical systems, I can mention the population of predator and prey year-over-year, the iterations of the Newton method, the gradient descent on a neural network…

In more technical terms, a continuous dynamical system is defined as a differential equation[math] dx/dt = f(x)[/math], while a discrete dynamical system is defined as iterations of a function [math]x_{n+1} = f(x_n)[/math].

One of Poincaré’s insights was that continuous and discrete dynamical systems are intimately linked. If you have a continuous dynamical system in an n-dimensional space, you can consider an n-1 dimensional surface [math]S[/math] (actually, it’s more generally called a manifold, surfaces are usually 2-dimensional), and consider the return map (now called the Poincaré map) to [math]S[/math], i.e. the first point of the orbit of [math]x[/math] that intersects [math]S[/math] again.

Much of the essence of the original, n-dimensional, continuous dynamical system is actually captured in this n-1 dimensional, discrete dynamical system. For instance, if an orbit is periodic in the original system, it’ll also be periodic in the Poincaré map. If another orbit converges to that periodic orbit, it also will in the map, as it will also be repelled on the map if it’s repelled in the original system.

With that, most of the work now can be done on a simpler system, and the results can be applied to the complicated differential equations we began with. So, for the rest of our story we’ll focus only on discrete dynamical systems, i.e., iteration of bijective differentiable functions (actually we usually require that the inverse function is also differentiable, i.e., that the map is a diffeomorphism) of a space in itself, [math]f: M \rightarrow M[/math].

Now, suppose we have a periodic point of [math]f[/math], i.e. a point [math]P[/math] for which [math]f^n(P) = P[/math], for some [math]n>0[/math]. Replacing [math]f[/math] with [math]f^n[/math], if necessary, we can assume, to simplify, that [math]P[/math] is a fixed point, i.e. [math]f(P) =P[/math].

There are three possibilities for this point (excluding degenerate, unstable, cases with Lyapounov exponents zero):

• All points sufficiently close to [math]P[/math] get attracted to [math]P[/math] as the time goes forward, i.e., [math]f^n(x) \rightarrow P[/math]. In this case, we call [math]P[/math] a sink, or attactor.
• All points sufficiently close to P get repelled by P as the time goes forward, or, equivalently, they get attracted by P as time goes backward, i.e. [math]f^{-n}(x) \rightarrow P[/math]. In this case, we call [math]P[/math] a source.
• Neither of those are true, but there are some points that get attracted to [math]P[/math] and points that get repelled by [math]P[/math], no matter how close we get to [math]P.[/math] In this case, we call [math]P[/math] a saddle.

To see why these points are called saddle, imagine the our system is defined by a particle on some kind of field that generates potential energy, and the system tends to the minimum energy configuration, for instance, a ball rolling down a rugged terrain. A source in this case would be a peak, a local maximum of the height (energy), a sink would be a valley, a local minimum. A saddle, on the other hand, would look like the picture below.

To understand the dynamics around this saddle point, let’s look very close to it. Remember that we assumed that our map [math]f [/math]was differentiable, so, very near [math]P[/math], [math]f[/math] will look very much like a linear map, namely the derivative of [math]f[/math] at [math]P[/math]. Being a linear map, we can understand its behavior looking at its eigenvectors - the directions that keep fixed under the map - and eigenvalues - how much the fixed directions are expanded or contracted by the map.

If you know a bit of linear algebra, you can see that a fixed point is a sink when all the eigenvalues are less than 1 (or have norm less than 1, if they’re complex) - i.e. the map contracts all directions -, and a source when all the eigenvalues are greater than 1. A saddle point, on the other hand, has some eigenvalues greater than 1, and some smaller.

Let’s look at the simpler possible saddle point, in two dimensions - one expanding (>1) eigenvalue, and one contracting (<1).

Very near P, if we draw, in blue, the points that get attracted to [math]P[/math] as time move forward, [math]f^n(x) \rightarrow P[/math] - we call it the stable manifold of [math]P[/math] - it will be very close to a straight line, on the direction of the contracting eigenvector. Similarly, if we draw, in orange, the points that get attracted to [math]P[/math] as time moves backwards, [math]f^{-n}(x) \rightarrow P[/math] - the unstable manifold -, it will be very close to the direction of the expanding eigenvector.

However, if we keep drawing the stable and unstable manifolds farther away from [math]P[/math], thing will deviate more from the linear map, and nothing prevent them to curving away from their initial directions.

What Poincaré noticed was that, on the n-body system, sometimes the stable and unstable manifolds curved so much that they crossed each other!

This point [math]Q[/math] is called a homoclinic point, and Poincaré showed that, when this occurs, a lot of strangeness start to appear in the system. The first thing to notice is that, by the definition of the stable and unstable manifolds, the orbit of [math]Q[/math] will converge to [math]P[/math] both in the future and in the past! More than that, it also means that every point in [math]Q[/math]’s orbit will share this property, i.e., will be in both manifolds (i.e. be homoclinic points)! Now, try to picture what needs to happen with the blue and orange lines so that [math]f(Q)[/math] and [math]f^-1(Q)[/math] are also intersection points.

The only way for this to happen is if both lines keep folding over themselves, and that will continue happening for the whole orbit, i.e. infinite times. A few more iterations, and our picute will start looking like this.

By this point, I guess you’re starting to see where the fractals may enter our story, but before that, let me introduce our second hero.

Enter Stephen Smale.

If you don’t believe in fate, Smale’s career can start challenging your beliefs. It’s hard not to believe that his mission on Earth was continuing Poincaré’s legacy. It all started when, in 1961, he proved one the most famous open problems in topology at the time, that any n-dimensional topological space that’s simple enough necessarily needs to the homeomorphic to the n-dimensional sphere (he proved it for n>4, the cases n=4 and n=3 were later proved by Freedman and Perelman, respectively. All three of them received Fields Medals for that). By now, you may have guessed who stated that conjecture, in 1904. Yes, Henri Poincaré!

After being catapulted to the pinnacles of mathematical glory by the proof of the Poincaré conjecture, Smale decided to work on other of the great mathematical branches planted by Poincaré - dynamical systems.

And the next act of our story happens here, in Copacabana beach, Rio de Janeiro.

For most of the 1960’s, Smale spent the summers in Rio, collaborating with local mathematicians and, of course, escaping the freezing cold of his natal Michigan. (Actually, he kept the habit of going frequently to Rio until much later. I remember seeing him on a public bus once there in the 90’s when I was a graduate student, and wondering if anyone else had any idea that that nice old man sitting on one of the back seats was simply one of the most intelligent people alive)

As they loved to say, it was on the sands of Copacabana that one of his most famous ideas came to him - the Smale horseshoe map.

It’s one of the ideas that, in hindsight, seems so obvious that makes you wonder why nobody had really thought this before. Look again at our saddle point.

Now draw the blue rectangle, and think what [math]f[/math] will do with it. First, [math]f[/math] wil contract it horizontally and strecth it vertically, in the directions of the contracting and expading eigenvectors, respectively. But then, it will curve along the unstable manifold, folding on itself, and, maybe after some iterations, it’ll become something like the orange “horseshoe” in the picture.

Like Poincaré did with the return map, now Smale again turned the problem of understanding the system into the simpler problem of understanding the simpler horseshoe map - if we understand this simple map of the rectangle, we’ll understand much of the dynamics of the homoclinic intersection!

But the first thing to notice is that the horseshoe map isn’t a map of the rectangle on itself - it curves outside the rectangle, and then intersects it again only on the colored regions below.

So, if we want to understand how the stable manifold folds on itself, we can observe that (maybe having to replace [math]f[/math] with [math]f^n[/math], for some [math]n[/math]), the points on the stable manifold converges to P, which means that they stay in the blue rectangle. But what are the points [math]x[/math] in the rectangle such that [math]f(x)[/math] is also in the rectangle, i.e., in one of the colored areas?

If you play back in your head what the function is doing, and unbends the horseshoe, it isn’t hard to see that those points form two horizontal strips.

Now, let’s find the points [math]x[/math], for which [math]x[/math], [math]f(x)[/math] and [math]f^2(x)[/math] are all in the rectangle. Now, besides being in the original colored areas, [math]f(x)[/math] also has to be inside the intersection of the horizontal strips with them. Again, if you play the movie back on your head, you’ll see that now we’ll have two strips within each of the initial strips!

I think now it wouldn’t be a surprise if I tell you that the set of [math]x[/math] such that [math]x, f(x), f^2(x)[/math] and [math]f^3(x)[/math] are all in the rectangle is the union of 8 horizontal stripes, two in each of the four stripes above.

Hmmm… I think I’ve seen this before. Where did we start with an interval, then selected two subintervals, then two subintervals of each of them, and so on, ad infinitum? What did we get if we repeated that infinite times? Yes, a Cantor set!

So, the set of points that will remain in the rectangle forever in the future is a Cantor set of horizontal lines! And, applying the same reasoning, the set of points that will remain in the rectangle forever in the past is a Cantor set of vertical lines.

If we go back to our original problems, that means that both the stable and the unstable manifolds are folding on themselves forming Cantor sets!

So, we started with very natural problems - the stability of the Solar System and the dynamics around saddle points -, and naturally we found fractal structures at the core of the system’s behavior.

But wait, there’s more! Ok, you say, there’s a fractal there, that’s cool; but I bet that BS about non-integer dimensions is just a cool useless thing - no way that’s useful in practice!

We’ll need a bit more work until we get there, but I don’t want to start the third act without introducing our third hero - Jean-Christophe Yoccoz.

Yoccoz was probably the sharpest mind I’ve had the pleasure to talk to. I remember once spending more than one week on a problem, and, being stuck at a particular point, going to talk to him. When I finished stating the problem, before I even started to explain the progress I’d made, he had already retraced all the steps I’d done, solved the issue I was stuck on, and generalized the result. All of that in less than five minutes!

But as I said, we’ll get to him in a minute. Let’s first go back to our homoclinic point.

The homoclinic point I drew, as far as homoclinic intersections go, was reasonably well behaved. This is because it is a transversal homoclinic intersection, i.e. the stable and unstable manifolds cross each other at a positive angle. When all homoclinic (and heteroclinic, between the stable manifold of one periodic point and the unstable manifold of another) intersections are transversal, one can prove that the system displays some degree of stability. When the manifolds intersect tangentially (i.e. at angle zero), all hell breaks loose.

When that happens, we call it a homoclinic bifurcation. It’s reasonably easy to understand that, if we perturb our map a little, say by changing it to [math]f_\epsilon(x) = f(x) + \epsilon g(x)[/math], with [math]\epsilon[/math] small, the stable and unstable manifolds will move a little. So, if they meet at an angle [math]\alpha > 0[/math], now they’ll still meet, at an angle close to [math]\alpha[/math]. This is why, for intersections to appear (or disappear), they need to go through a tangency.

Now, assume we already have a transversal homoclinic point, and we keep deforming the map, until it has, besides this point, also a tangency. Near the tangency, the stable and unstable manifolds will look like this.

Remember that both the stable and the unstable manifolds are (locally) Cantor sets of lines.

We can always change the coordenates locally to make the stable manifold look perfectly horizontal, and the unstable manifold to have their horizontal tangents exactly at the dotted line. So, if we look at the intersections of the manifolds with the dotted line as we perturb our function, what we’ll see are two Cantor sets “moving through each other”, and, whenever they intersect, we have a new tangency!

So, essentially, we have two Cantor sets K_1 and K_2, and we want to find the t such that K_1 intersects K_2 + t, or in other words, the set ot t for which there exists x_1 \in K_1 and x_2 \in K_2, such that x_1 = x_2 + t, i.e. t = x_1 - x_2. So, in other words, the set {[math]x_1 - x_2| x_1 \in K_1, x_2 \in K_2[/math]}.

A good way to visualize the set of the differences between two sets of real numbers is the one below.

You start building the cartesian product of the two sets, and then you project them back on the horizontal axis, using a 45-degree projection. It’s easy to see that each of the triangles composed by a blue point, a red point and a green point are isosceles and that this implies that the x-coordinate of the green point is the difference of the x-coordinate and the y-coordinate of the red point.

If we apply this method to our Cantor sets, our set of the t’s such that a new tangency is created is the projection of the cartesian product onto the real line. If we think now on how the cartesian product work with our regular, integer-dimension objects, we’ll find two interesting properties:

• dim (A x B) = dim(A) + dim (B)
• If we project a set on a set with smaller dimension, it’ll have the full dimension of the target set, except on very degenerate cases. For instance, a 3-d object usually cast a 2-dimensional shadow on a 2-dimensional wall, and it usually covers positive area (i.e. it contains full squares, or disks).

It turns out that the same is true for fractional-dimensional Cantor sets. If you have a 0.7-dimensional Cantor set [math]K_1[/math] and a 0.6-dimensional Cantor set [math]K_2[/math], the cartesian product [math]K_1 \times K_2[/math] will have dimension 1.3, and its projection to the real line, except on degenerate cases, will be 1-dimensional, and will contain full intervals.

And that is the essence of Yoccoz’s result. If you perturb a homoclinic tangency whose stable manifold’s and unstable manifold’s cross-section dimensions sum up more than 1, there will be intervals for which, on any point on that interval, a homoclinic biffurcation is happening!

And there are several prior theorems that say that, when that happens, all sorts of really weird things happen in the system - infinitely many periodic attractors, infinitely many strange (fractal) attractors, and many other odd behaviors.

So, in a nutshell, we’ve seen that fractals and fractal dimensions are at the very core of the study of dynamical systems, and are intimately linked to the stability (or lack thereof) of the system’s dynamics.

Cellular automata are used to model natural phenomena in almost every discipline. Despite their simplicity, they are capable to solve many intractable scientific problems.

They are used in

1. Physical systems: One of their applications is description of recrystallization in certain types of metals an alloys. They describe dendrites during crystallization.
2. Description of Percolation within structures.
3. Cracks propagation within materials.
4. Laser behaviour is modeled.
5. Avalanches. Think about Per Bak sand pile models.
6. Traffic flow on highways and within cities.
7. Description of Flood propagation within the country.
8. Biology to model ecosystems
9. Chemistry: chemical and biochemical reactions.
10. Medicine, cellular automata model spread of diseases.
11. Coding messages by cellular automata, which produce very strong codes.
12. Picture compression gives very good results.

Descendants of cellular automata and lattice gasses are called lattice Boltzmann models describe flows in:

1. Medicine: flow of blood through vessels, their interactions with vessel calls, simulation and prediction of pressure within aneurysm, etc.
2. Physical systems: droplet behavior (ink printer, etc.)
3. Wetting of surfaces.
4. Liquid versus deformable solid interactions.
5. And many other apps.

Cellular automata can be applied above another types of the computational worlds and not only above the lattices. For example, they can work above networks.

It is important to mention that cellular automata are often part of multiscale models, where they usually work at the mesoscale levels.

They enables theoretical studies of self-organization and emergence. They are extremely well suited for theoretical studies of complex systems per se.

Cellular automata serve as an easy to understand test bed of more advanced models and often solves the problem under consideration with sufficient preciseness. They are a perfect theoretical tool.

What a wonderful way to phrase this question! It makes itself so simple to answer - as if everything that could be said in response is already contained within the query…

This is why I love to say ‘the proof’s in the question’. If fractals can be described at all they must therefore exist in an environment with fractal properties (such that ‘orders of magnitude’ can be described in a fashion parallel to the interactions of the geometric function of a fractal). In simpler terms, for us to codify ‘fractals’ those codes themselves must exist in a logical space whose own repetitive self-expositions allow ‘levels’ to be differentiated. As an even more simplistic rendering: “reality is a fractal”. Most basically… “existence exists”.

This fact’s depth is in its superficiality and its most fundamental function is universal.

This is what makes fractals fascinating - they are the abstractions that mirror actuality in a manner that exposes abstraction as abstract and actuality as actual.

Where it gets really fun is when we allow dimensionality to be a fractal function itself

Do cellular automata disprove the teleological argument.

No. In order for cellular automata to disprove the teleological argument, it would be necessary to have clarification of the Mind/Body Problem (MBP). In that regard, no natural phenomenon is sufficient for a disproof of a supernatural phenomenon.

But why? Science itself, whether Biological, Physical, Chemical, or otherwise, need clarification as to whether or not it is premised upon metaphysical or supernatural terminology or whether or not it is premised entirely upon natural or physical terminology . Abstract Conceptions have never been shown to exist in a causal relation to natural phenomena. “Disprove” itself, is an abstract conception and a supernatural phenomenon (metaphysical phenomenon) as is the identity “cellular automata”)

This intermingling of concepts which are categorically distinct exhibits the essential limitation of reductionist thinking. What this means is that mental identities are sometimes used as though they were equal to the same attributes of body identities. That is a logical incoherency, or what is referred to as a violation of the Aristotelian Law of Noncontradiction.

Notwithstanding, Philosophical Behaviorism has been advocated by psychologist B. F. Skinner and the reductionist thinking used produced the kind of “Meat—Machine” or “Robot Vehicle” for people as a primate species. The scientific definition for “human” is “primate”. Most people do not realize this, because all words associated with “human” are misunderstood to be representative of some identity equivalent to a kind of “Self” or a unique “person” which is endowed with some ethical conceptions of “rights”,. This is problematic because nowhere in the anatomy of the Meat-Machine Model or Robot Vehicle Model, , has a “self” or “person” or “right” ever been found. Neither has any scientist been credited with the discovery a “self” or “person”.

Further considerations which go far deeper, involve recognizing that all operations of “law” in any Justice System, intended to punish crimes, are not at all logically coherent with Philosophical Behaviorism. Any citizen charged with a crime, can reasonably offer testimony denying their responsibility for their actions, because it can be argued that:

(a) “The Biology Did It” or (b) “The Chemistry Did It”

or (c) “Nature did it, as all human functions are Natural, and therefore conceptions of crime or unlawful behavior cannot apply to ME! Everything I have done corresponds with the “Laws of Nature”.

In regard to this, it can be helpful to understand the Violence of Nature Theory, as articulated by Robert Sapolsky. Robert Sapolsky, Ph. D. (Endocrinology). Doctor Sapolsky holds a professorships in biology, neurological sciences and neurosurgery at Stanford University.

"Lots of other species kill. They kill members of their own species. The kill them in anger. They kill them in cold blood, premeditation. They kill them strategically, In ways that would make Machiavelli proud. They do it to each other's infants. They do it competitively, advantageously. We're not the only species that kills."-------from the "Class Day Celebration Speech" by Dr. Robert Sapolsky, professor of biology, and professor of neurology and neurological sciences and, by courtesy, neurosurgery, at Stanford

http://University.polskyhttp://youtu.be/hrCVu25wQ5s

Philosophical Behaviorism lacks explanatory power and behaviorism is largely dismissed in the 21st Century for these glaring insufficiencies, although it still has an advocacy with researchers with materialistic and reductionist leanings.

SUMMARY:

Most of the world’s population is not familiar with these conceptions, and as a result, they have a rather benign array of thoughts about science, as though behavioral science somehow supports conceptions about “human good” and the wondrous and beautiful conceptions of a unique individual self. That is not the case.

What Doctor Sapolsky exhibits is selective logic, or what is called an ARGUED SUBJECTIVITY. This is the logical error supposing that the ideas held actually explain how the individual acts and operates, whereas in fact, the individual, neuroscientist or otherwise, lives continually in the socially comforting atmosphere wherein one does not expect to be murdered and their body cooked for another person’s dinner. It is a way of theorizing about life as actually being quite “brutish” while knowing with assurances, that the 7 PM coctail party at the university facilities will be pleasant and cheerful, with police protection only minutes away at all times.

The conceptions of the Robot Vehicle are not favorable to individual rights at law called “human rights” or other conceptions about the existence of an individual “Self” that is worthy of empathetic feelings and “Human Interest Stories” in the mainstream media. This is not however, logically coherent with what Dr. Sapolsky intends to reveal. Philosophical Behaviorism explains only that mankind are the inheritors of the most brutal mentalities and violent urges currently associated with terrorist organizations which engage in the bombing and genocidal attacks on civilian populations.

According to the Robot Vehicle model of what is called the “human being” all that can be reasonably expected of humanity is raw barbarism in a universe of randomness that is devoid of all ethical foundations..

See also: Philosophical Determinism and B. F. Skinner

In this universe, no

For the simple fact that this universe is finite.

“Yes but what about if it is infinitely small”

No, once you reach the atomic level, it becomes impossible to keep repeating the structures.

A Mandelbrot fractal zoom of over 1511 magnitudes

The diameter of the known observable universe is [math]8.8\cdot10^{26}m[/math]. From that, down to 1 meter, is is nearly 27 magnitudes (powers of 10).

The hydrogen atom has a diameter of about [math]10^{-8} m[/math]… another 8 magnitudes.

The smallest physical distance that makes any sort of sense (as of 2020) is the Planck Length, at [math]1.62\cdot10^{-35} m[/math].

So, the difference between the largest conceivable distance to the smallest, is roughly 61 magnitudes.

The video above breezes through that in under 5 minutes.

…then keeps going for another 1 hour and 52 minutes.

No, physical fractals are very definitely finite. At least in this universe.

The universe is a fractal. Gravity dissipates with inverse square, too much gravity is a black hole which can't get more dense in kg/m2, it just gets bigger, so the amount of gravity in the universe is at most some constant it's radius squared. Everything we know(quasi particles don't count) has positive gravity, so the amount of stuff in the universe is at most radius squared, In black holes it's exactly radius squared. The universe is the same or less dense than infinite paper(topological and Hausdorff 2dimensional), so it's Hausdorff dimension is 2 or less, but it's topological dimension is 3, different topological dimension from Hausdorff dimension is fractal.

Obviously subjective and way too open-ended of a question … but since you asked:

1. The best cellular automata using a simple totalistic rule on a square grid using the 8-cell neighborhood is Conway’s Game of Life.
2. The best cellular automata using a simple totalistic rule on a square grid using the 4-cell neighborhood is unknown. Or at least I don’t know of a good one…
3. The best cellular automata using a simple totalistic rule on a hex grid using the natural 6-cell neighborhood is my creation/discovery Joe Life described here.
4. The best cellular automata using a simple totalistic rule on a triangular grid is this one, again described by me here — although in this case I’m not sure that I didn’t rediscover it. It is clearly the best given the 12-cell neighborhood and 2 states on a triangular grid and the 3-cell neighborhood doesn’t support anything interesting using only 2 states.

The theory of consciousness units (CUs) and cellular automata both tap into fundamental mechanisms for complexity and emergence. Here’s an exploration of how these ideas might intersect:

Consciousness Units (CUs): The idea posits that consciousness exists in discrete units of intelligent energy, similar to fundamental particles in physics but applicable to conscious experience and intent. CUs may combine or interact to give rise to larger, complex forms (gestalts) of consciousness or even influence physical reality, especially when combined with focused intention.

Cellular Automata: This mathematical model consists of simple, discrete cells that evolve over time based on predefined rules. Each cell interacts only with its immediate neighbors, yet together, they can generate complex, often surprising patterns.

Both CUs and cellular automata represent a system of units with local interactions that collectively lead to higher-order complexity. CUs might act as fundamental “cells” of consciousness, each with a simple set of properties or rules, where the interaction of these units gives rise to complex conscious phenomena or intentions. In cellular automata, complex behaviors emerge from simple rules without a central control mechanism. If CUs follow similar principles, then higher states of awareness or spiritual manifestation could emerge from the interactions of CUs, rather than from a single “central” source. This aligns with the idea that consciousness, like life in cellular automata, emerges from a foundational substrate.

In cellular automata, rules guide each cell’s state based on the state of neighboring cells. For CUs, we might envision rules where units respond to surrounding CUs, emotional states, or intentional focus. When enough CUs align or “focus” on a particular intention, they could manifest change on a physical or energetic level, analogous to how patterns stabilize or evolve in cellular automata. Cellular automata models could theoretically simulate CU interactions by assigning each “cell” a basic state of consciousness, intention, or emotion, and observing how these states influence each other under specific rules. The results could help illustrate how small units of intention interact to create a conscious experience or even cause physical manifestations, in line with the CU theory.

If physical reality emerges from the arrangement and interaction of consciousness units, then each manifestation could be seen as a unique configuration of these units. Cellular automata demonstrate how even simple, iterative interactions can lead to dynamic patterns, hinting that the material world might similarly be a dynamic arrangement of underlying CUs, brought into form by their cumulative states and interactions.

In essence, just as cellular automata evolve into unexpected complexity from simple rules, consciousness units could similarly form the complex fabric of consciousness and reality by combining and interacting according to foundational principles. This analogy opens avenues for exploring consciousness and intentionality through simulations, potentially yielding insights into how reality might be influenced or “shaped” by collective focus and intent.

"A New Kind of Science" by Stephen Wolfram is a good reference book on cellular automata.

Welcome to Fractal Land!!!

What are Fractals? A fractal can be defined as a mathematical set exhibiting a repeating structure or a pattern displayed at every scale, also known as expanding symmetry. An object is called a self-similar one if the repetition is same at each scale. A famous example of such a pattern is the Mandelbrot set itself which gained popularity because of its aesthetic charisma. Magnifying or zooming an image of a Mandelbrot set reveals its self-repeating properties.

The word ‘fractal’ was coined by Benoit Mandelbrot and this word became popular within a short span of time. The idea of the word ‘fractal’ was taken from the Latin word ‘fractus’ which means ‘to create irregular objects’. These concepts of fractals, irregularities in objects, self-similarities, patterns attracted artists all over the world. This resulted in a genesis of what we know as the Fractal Art. Researchers from various domains related to Signal Processing and Composition started using the idea of fractal dimension in their works. New algorithms developed including the famous Higuchi Fractal Dimension and Katz Dimension to exploit the fractal dimension in Audio Signals. This idea then extended to analyze the business trends, social trends to extract information regarding whether there will be a rise or fall in the sales.

The foundation of Chaos Theory inspired mathematicians to study the nature-to unveil the Nature. Nature has ample examples that needs to be studied and Chaos Theory accompanied with this fresh concept of Fractals be major tools for the scientists to conduct those researches. We are aware of the butterfly effect-most of us know about this phenomenon because of the movie of the same name. Nevertheless, it is an exciting topic-the underlying concept is Chaos. The weather system, weather forecast all of these can be studied, the underlying principle being Chaos Theory. The aim is to find pattern in these Chaos. This is the idea of Fractals-to find regularities in the existing irregularities. Mathematics inspired artists from time to time. The Euclidean geometry, the concept of Fibonacci Sequence and the Golden Ratio attracted painters and sculptors. This fact is evident from their creations. In the recent past the artists are imbued with this newly emerging concept of Fractals and these are reflected in their arts. Digital media has progressed a lot in this respect.

The concept of dream within a dream and within a dream in the movie ‘Inception’ was nothing but a repeating structure-the concept of Fractals. What we find in nature is in a way has some form of fractal dimension in it. Good examples of fractals found in nature are broccoli, cauliflowers and so on. Careful observation into the structure of broccoli reveals its self-repeating properties. Same applies for cauliflowers as well. These are nothing but fractals. A leaf has self-repeating patters, our skin, fish scales-all of these are fractals. An interesting experiment can be done like this. Let us consider a simple structure. If we go on repeating it eventually it will lead to a different structure-a bigger one, a better one with a more aesthetic appeal. This is how music is made. At a very small scale if we observe a music signal we can get repeating patterns of the same signal. A music which seems pleasing to our ears, that is has a more aesthetic value has more fractality than that of one which has less aesthetic value. A simple comparison in this regard can be made by considering to music signals like a simple noise and a piano cover of Bohemian Rhapsody. Which sounds more pleasing? Of course the piano cover sounds more pleasing. Studies show that this simple fact can be shown by comparing the Hurst exponent values-a parameter for judging self-similarity in a signal; the piano cover had a Hurst exponent value much greater than that of the noise suggesting the same as judged by our ears.

Scientific ventures into studying the emotions associated with Audio signals, both Music and Speech signals have been made recently and the results came out to be satisfactory. What a layman thinks as a music associated with sad emotion or happy emotion can be identified properly with these approaches involving the estimation of the fractal dimensions.

Ever noticed the temples? All of them incorporate self-repeating structures. This idea was innate in the artists but then there was little scope for mathematical interpretation. The Eiffel Tower looks fascinating only because of the existence of fractality in it. Bridges have fractality in the structures. This domain is called Fractal Architecture. That which is not fractal has no aesthetic value, been regarded as noise and we discard them. Nature approves of it-so most of the things existing in nature, including the man-made things are fractals.

A drop of ink when placed on a piece of moist paper when gets soaked takes a shape which is self-repeating. The branches of a tree are self-repeating, nothing but fractals. The famous paintings of artists like Jackson Pollock also have fractal elements in it. One of the examples can be The Autumn Rhythm. Recent advancements in the field of bio-medical research works have been made with the concept of fractals. Recent studies have been conducted with famous literature of eminent authors. The idea is to check the frequency of occurrences of words in a span of say around hundred pages or an entire composition. By this approach we get a histogram which will have a pattern, we need to identify the fractal dimension and the associated emotion of the word that is reflected in the composition. These approaches can be applied to lyrics of a song as well.

The Nature provides plentiful examples. Fractals have wide applications in Physics, in Signal Processing, in Cognitive Neuroscience and in Psychology, in Mathematics, in Arts, in Social Sciences as well. It is one of the interesting things existing in nature-The Chaos Theory.

This is a pictorial representation of the famous Mandelbrot Set.

I do apologize if this response to your question sounds a little tongue-in-cheek.

It sounds like you are asking whether it is necessary, or even useful to you, to learn about fractals. Many people live their lives quite happily, usefully and successfully while not being aware of the existence of the concept of a “fractional dimensional reality” - or “fractal reality”. Nevertheless that is the kind of reality in which they live.

Ideas do not have to have a “real life application” in order to be true models of the real world. Virtually everything is fractal in nature. But, can a person earn a living using knowledge of this fact, you ask? I think yes. In fact, some people are getting filthy rich using knowledge of fractals.

For example, most computer-generated images (CGI) are created using knowledge of fractals. But, I don’t think that I would consider the making of modern fictional movies a “real life application”. It is more of a fantasy-life application.

One of my favorite fractals is called the Harter-Heighway curve ( Dragon curve - Wikipedia ) It is also called the dragon curve, or the “paper-folding curve”. If you have a piece of paper on your desk, you have a potential dragon curve. Take that piece of paper and fold it four times, then unfold it partially. You have a dragon curve.

Not all cellular automata are Turing-complete, but some are. A cellular automaton is Turing-complete if it can implement any Turing machine (or, equivalently, implement a universal Turing machine.)

The usual method to demonstrate is the same as with programming languages: explicitly create a Turing machine, or some other construction already shown to be equivalent. See How do you prove that a programming language is turing-complete? or What does it mean to say that a programming language is Turing Complete?

The fact that 2-D and 1-D cellular automata are capable of universal computation has been known since at least 1971. See http://alvyray.com/Papers/CA/UniversalCA_1D.pdf

Given a Turing machine, we can create a 1-dimensional CA whose states are all combinations of “tape with symbol X” or “tape with symbol X, with Turing machine head present in state Y” . Then the transition rules can easily implement the Turing machine, by translating the Turing machine’s state transitions and actions. This leads to a large number of possible cell states, so one area of research has been demonstrating Turing-completeness with fewer states and rules than the “literal” translation.

For Conway’s Life, the original proof of universality showed that it was possible to create signals (using gliders) and combine those signals with conventional logical gates such as AND, OR, and NOT. The proof also demonstrated that the position of a block could be manipulated and measured with gliders, to create an unbounded storage element. These two elements are sufficient to build a digital computer inside Conway’s Life, which is known to be equivalent to a Turing machine (Subsequently, people have created much more compact demonstrations of computation within Conway’s Life— actually implementing designs such as a cellular automata running on top of Life computing elements.) Conway’s proof also demonstrated that is was possible for such a computer to erase itself, meaning that the problem of “will this cellular automaton ever reach population zero” is an undecidable problem.

Cellular automata are sometimes used for level generation in games, or as a procedural generation technique in other contexts, or as a lightweight physics model in games.

You can find numerous examples of the use of CAs for level generation explained in articles and tutorials; here’s RogueBasin’s version: Cellular Automata Method for Generating Random Cave-Like Levels

And, just for fun, a couple research papers in the topic!

• Cellular automata for real-time generation of infinite cave levels.
• Improving procedural 2D map Generation based on multi-layered cellular automata and Hilbert curves

Games such as Factorio or Dwarf Fortress are sort of like cellular automata, in that they divide the world into a fixed-sized grid, where most effects are local to the neighbors. But both have long-distance effects too, so are not pure CAs. It is common to see environmental or atmospheric effects (like diffusion) modeled using cellular automata.

Nope it's a stretching out wormhole phenomena .in a way but in another way it's fluid that has a solid container that when you introduce something inside it puts pressure on the container and causes pressure on what ever you introduce in the form of curvature , I don't know how to explain it but space has density that reacts to the amount of density of the object trying to take up space the more dense the object the greater the density of the curvature causing greater acceleration I guess. Another words I really am not sure . But ultimate density causes blackholes to form and greater than that causes black holes to explode ( big bang ) another words space time can maintain its integrity better than energy mass can

Hi. Below is a list of real-life applications with links to concise details. Vernon Goodman already mentioned image compression - the Yale link below provides additional information.

Nathan Cohen pioneered the development of fractal antennas:

Michael Barnsley pioneered the process of fractal image compression:

In addition, there are fractal-based capacitors:

fiber optics:

and mixers:

One of my favorite is invisibility cloaks. These were developed by the same Nathan Cohen who first experimented with fractal antenna design:

Here’s his patent for “Wideband electromagnetic cloaking systems”:

Fractal analysis is used to diagnose and treat liver disease:

and has potential uses in the diagnosis of heart disease:

This list is a small sample, but it underscores the profound ways in which fractal geometry has influenced our everyday lives.

You can definitely express cellular automata as matrix multiplication over finite rings. You do the same thing you do with a real-valued matrix to run a video filter, like blur or edge detection, only you evaluate the sums and products in somewhat arbitrary ways, rather than in R or C.

You have a square matrix of coefficients and you take the square matrix centered around each pixel and do a standard matrix multiplication, with nonstandard definitions of sum and product to get the next value of that pixel.

There is a lot more to matrix math than a simple evaluation procedure, from row elimination to vector fields, to eigenvalues. You can definitely compute these with carefully chosen cellular automata, because many CAs have patterns that qualify as simple computers, from the glider gun lambda calculus unifier in Conway’ Game of Life to Von Neumann's “Swarm” of self-reproducing 3D images with arithmetic specialties. But describing a general-purpose cellular computer is too complex to give in a quora answer.

Suffice it to say that a UTM is a UTM, there are several known cellular UTMs and they can do Linear Algebra.

This question seems to confuse a few concepts.

Rule 110 is a cellular automaton which, given some input, if one plots successive states, exhibits some chaotic patterns. The cellular automaton also has been proven to be Turing complete.

Turing-completeness is a property of any system which manipulates data in some way: programming languages, cellular automata, ... Specifically, it means that the system can, given the right input, perform the computation equivalent to any Turing machine.
To prove that a system is Turing complete one needs to have a way to convert a problem to an appropriate input, have a way to determine when the computation has terminated, and a way to interpret the output.

A fractal is any object which has self-similarity: parts of it are the same or highly similar to the original. Rule 110 started with a single cell set to 1 leads to an interesting pattern, which could arguably be said to be fractal-like (although other rules in the same class generate the Sierpinski Triangle, one of the most typical examples of a fractal, rule 110 generates a pattern somewhere in between structure and chaos).

For a general fractal, it is meaningless to ask whether it is Turing complete.

Computer processors
Cellular automaton processors are physical implementations of CA concepts, which can process information computationally. Processing elements are arranged in a regular grid of identical cells. The grid is usually a square tiling, or tessellation, of two or three dimensions; other tilings are possible, but not yet used. Cell states are determined only by interactions with adjacent neighbor cells. No means exists to communicate directly with cells farther away. One such cellular automaton processor array configuration is the systolic array. Cell interaction can be via electric charge, magnetism, vibration (phonons at quantum scales), or any other physically useful means. This can be done in several ways so no wires are needed between any elements. This is very unlike processors used in most computers today, von Neumann designs, which are divided into sections with elements that can communicate with distant elements over wires.
Cryptography
Rule 30 was originally suggested as a possible Block cipher for use in cryptography. Two dimensional cellular automata are used for random number generation.
Cellular automata have been proposed for public key cryptography. The one-way function is the evolution of a finite CA whose inverse is believed to be hard to find. Given the rule, anyone can easily calculate future states, but it appears to be very difficult to calculate previous states. However, the designer of the rule can create it in such a way as to be able to easily invert it. Therefore, it is apparently a trapdoor function, and can be used as a public-key cryptosystem. The security of such systems is not currently known.
Error correction coding
CA have been applied to design error correction codes in the paper "Design of CAECC – Cellular Automata Based Error Correcting Code", by D. Roy Chowdhury, S. Basu, I. Sen Gupta, P. Pal Chaudhuri. The paper defines a new scheme of building SEC-DED codes using CA, and also reports a fast hardware decoder for the code.

I've read A New Kind of Science by Dr Stephen Wolfram. Wolfram is a brilliant individual, and should be heard out. In his book, he uses the idea of using Cellular Automata to develop physics, instead of using the laws of physics. This was an intrigiung concept. But I have seen few applications of it by physicists.

In some ways, his thesis is that simulating reality with cellular automata is better than trying to model reality with equations. And maybe physics will get so complicate that the only thing we can do is to simulate it.

But for now, Wolfram's ideas do not seem to have sparked much interest.

CA are abstract mathematical models and not super “practical”. This isn’t very different from much of mathematics. But, studying CA can lead to insights which can be applied to more practical models.

There is this gap that exists in our minds, between the artificial (technological) systems that humans create and the natural (biological) systems that nature creates. But there doesn’t have to be a gap.

CA are artificial systems that can do things that seem “life-like” to us. This, along with other models, can help us come to the realization that we can create systems that are neither natural nor artificial, but something in between. This gives rise to fields such as artificial life and morphogenetic engineering.

Are there actual jobs in it?

Mostly mathematicians and computer scientists doing theoretical research. There might be some applied researchers/engineers out there that use CA incidentally while working on more practical projects. I can imagine an engineer working on a self-replicating robot might create CA simulations to gain a better understanding of the task.

More reading:

• Artificial life
• Morphogenetic engineering: http://www.lacl.fr/~michel/PUBLIS/2013/Doursat_Sayama_Michel_2013_MorphEng_review.pdf
• Complex systems

Good question, for which science is not yet able to provide anything close to a definitive answer. The reason I think it might be relevant is that many factors mutually interact in a nonlinear fashion, including, but not limited to:

• Mutation rates
• development
• Interplay of sexual vs other kind of selection
• Epigenetic factors

Now, I am not saying that fractal mathematics applies, per se, but that when such complex factors interact in nonlinear and complex ways, well, that is the kind of environment where chaos theory and fractal patterns are sometimes applicable. None has yet come up with a deep mathematics around evolution.

It may not be complete answer because it will become lengthiest explanation i have ever done if i explain (i am lazy :) :P :D) (one can have PhD on the topic)

I personally believe, yes it is. When i was in M.Tech 1st Year (2012-13) my automata Prof. Mr. Jaswinder SIngh gave me a topic "Stochastic Cellular Automata " for term paper. Initially i thought topic is of no use for me because my interest was in networking, but when i was doing my literature survey, i read about Chaos Theory,Butterfly effect, String Theory and others. Then i imagined it will be possible to simulate the whole universe in n-D Cellular Automata. I was amused by the concept. I know it will take a lot of computation power to simulate one of the real life scenario based on this model, But i still think it is possible. (I may have the answer or idea but that will trigger somethings that i will love to avoid)

Lets take an example, weather forecast are not accurate because they don't compute all the elements in the system. A tiniest detail can effect the whole system at large scale (as butterfly effect).

The thing i am saying n-D, because we live in 3-D space and we have to take time as another dimension, so it will be 4-D, and by studying more we can define other dimensions also for Cellular Automata. And about the elements and its states, we can reach beyond matter and Anti-matter, lets imagine for one second that String is our element in automata and its movements are states of a cell. We have to compute beyond 0 & 1 state of a cell. By computing avg of sector we can upscale a particular region and down-scaling will not be possible beyond string (let's say).

On the other it is also possible that Cellular Automata may be a representation of real universe, same as screen for images of real world. Then cell will be similar to pixel and color will be its state. In that concept we can think of 3-D or n-D representation of universe (as hologram). Timeline concept may be applicable too.

Fractals have been found to optimize flow of inputs and outputs in a manner that best allows for future growth.

• For biological systems, this explains why nutrients and cellular trash best flows through the fractal network called the vascular system, and the airflow pathways of the lungs.
• For plant systems, they have their own vascular system that does the same.
• For neural systems, this explains the fractal nature of neurons.
• For cities, this explains some of why roads have a somewhat self-similar nature too.

Meanwhile, in the non-biological world, the stochastic or “random-walk” nature of particles causes:

• The fractal pattern of beaches
• The fractal pattern of galaxy clustered in the universe
• The fractal patterns of mountains and streams

Additionally, the random-walk is likely the mechanism for the biological systems to optimize flow through the fractal nature.

Furthermore, just searching the self-similarity in general, one begins wondering about other self-similarities between seemingly non-related parts of the universe.

For instance, why do philosophical beliefs, existing in the mind, spread and divide in a way eerily reminiscent of evolution, which exists physically?

The answer is actually Game Theory. However, it brings up another question, why do completely unrelated phenomena all seem to have the same mathematical interpretations? Is this “related-interpretations” just another form of self-similarity?

In nature, I would say every biological living being can be considered to be a cellular automaton. We are all composed of cells, discrete blocks of which might closely or loosely interact with each other, depending on their specialization.

The discrete physical universe can be theoretically based upon cellular automata. However, if quantum effects are considered, that theory could break down.

Traffic modeling can be done using cellular Automata (albeit there are other methods).

Structural design can be done using CA by using lattices in a planar design and applying known forces models like wind, water waves etc to model the structure needed for building a bridge across a bay.

A cellular automata is a system where you have a set of cells, like the cells in a spreadsheet, that will change state based on the state of other cells. The most well-known is conways game of life. Your cells are arranged in a grid, and each is either occupied or empty. Each iteration, cells change state based on the following rules

1. Any live cell with fewer than two live neighbours dies, as if caused by under-population.
2. Any live cell with two or three live neighbours lives on to the next generation.
3. Any live cell with more than three live neighbours dies, as if by overcrowding.
4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

Depending on the initial condition, this can produce some very complex behaviors. You can have patterns of cells that oscilate in patters, roam across the grid, create other patterns, or remain stable. The simple set of rules can create a very rich set of behaviors.

There are many ways of implementing a cellular automata with different sets of rules, which will have different behaviors.

“Systems of nonlinear differential equations are a special case of an ex­tensive family of nonlinear dynamical systems into which also enter various nonlinear algebraic, difference, integral, functional and abstract operational equations. In this connection, until recently the uniform geometrical ap­proach to study nonlinear dynamical systems was represented absolutely natural, allowing to consider from common positions the nonlinear systems described by both discrete mappings, and ordinary and partial differential equations. Intensive application of geometrical approach to the analysis of dynamical systems originated with the well-known work of the American mathematician S. Smale who offered a design of mapping which subse­quently received the name of Smale's horseshoe. It has been shown, that stable limit set (attractor) of discrete dynamical system cannot be such a smooth manifold of the whole dimension which are, for example, a sta­ble limit cycle or torus, and everywhere holey, self-similar fractal set of fractional dimension. Besides it has been shown, that the behavior of tra­jectories of dynamical system for such strange attractor in the terminology of D. Ruelle and F. Takens is complex enough, combining global stability (the trajectory does not leave some area of phase space) with local instability of separate close trajectories, exponentially running up in time, that is characterized by the presence on an attractor both negative, and positive Lyapunov exponents. Other chaotic dynamical systems described by dis­crete mappings and possessing strange attractors have been further found, for example, logistic map, Henon map, Smale-Williams solenoid, etc.

As the analysis of properties of continuous dynamical systems described by ordinary differential equations, can be reduced, as it seemed, to the anal­ysis of properties of some mapping — Poincare mapping, it was observed that in continuous dynamical systems irregular, chaotic behaviour of tra­jectories became connected to the presence of a strange attractor in the system. However, the proof of this fact directly for the well-known Lorenz system of three ordinary differential equations in which the irregular behav­ior of trajectories for the first time was revealed, has faced with significant difficulties. For a long time, numerous attempts to prove by the methods of geometrical theory of dynamical systems the presence of a strange attractor in neighbourhoods of a saddle-node or a saddle-focus separatrix loops of the Lorenz system have ended with failure. Moreover, the problem to show, whether the behavior of solutions of the Lorenz system coincides with the dynamics of a geometrical Lorenz attractor was formulated by S. Smale as one of 18 most significant mathematical problems of XXI century. And the recent results of the authors have allowed to confirm definitely, that the geometrical approach developed for discrete mappings even if allowed to obtain a number of brilliant results for them, is not absolutely adequate with reference to the continuous dynamical systems described by the differ­ential equations. Now we can insist absolutely, that the definition of chaotic attractor of continuous dynamical system as a strange attractor, and also such traditional sections of chaotic dynamics as calculation of attractor's dimension, scenarios of transition to chaos, criteria of dynamical chaos and spatio-temporal chaos demand significant updating and revision.” (p. vii–viii)

World Scientific Series on Nonlinear Science Series A: Volume 58

New Methods for Chaotic Dynamics, Nikolai Alexandrovich Magnitskii.

https://doi.org/10.1142/6117

Yes certainly. Here are some practical uses:

1. counting things: you can’t count craters on the moon, or branches on a tree or rocks on a scree slope or trees in a forest without fractal methods. That is because the smaller things you count, the more you will find.
2. measuring things: you can’t measure the length of a coastline, the surface area of clouds or mountains without fractal methods. Again, the more detailed your measurement, the larger number you get. You need to use power laws, which is what fractal geometry is all about.
3. fractal geometry is often an efficient design. Fractal trusses have very high strength to weight ratio, a fractal cliff face dissipates wave energy really well, fractal tree structures conduct heat really efficiently etc.
4. fractal geometry is everywhere in nature: clouds, lightning, mountains, cracks, trees, forests, rivers etc. So understanding the principles is useful if you want to understand or do science in natural environments
5. fractal surfaces have very high surface area for their size, that can be useful for things like heat sinks, or as seeding (induce fizz in a drink, or as a place for coral to grow on), or to absorb sound.
6. fractals are attractive, and have been used in artwork and movies. Fractal art is a legitimate practical use

(high surface area fractal, from Exploring Scale Symmetry)

fractal heat exchanger (Oregon state university):