They are shapes that use up more space than their topology would suggest.

A Koch curve is topologically a curve, so one might expect it to have a one-dimensional coverage, but it actually is approximately 1.26 dimensional in its coverage of space.

The following surface is topologically like a plane, so ought to be 2D, but it occupies space in a 2.5D manner:

Fractal structures needn’t be simply connected like the above cases. These structures are tree-like:

apart from the right-most one, which is sponge-like.

Images from:

BRANCHING

Neurons from the human cortex. The branching of our brain cells creates the incredibly complex network that is responsible for all we perceive, imagine, remember.

Scale = 100 microns = 10-4 m.

Our lungs are branching fractals with a sur- face area ~100 m2. The similarity to a tree is significant, as lungs and trees both use their large surface areas to exchange oxygen and CO2. Scale = 30 cm

Oak tree, formed by a sprout branching, and then each of the branches branching again, etc. Scale = 30 m = 3*101 m.

SPIRALS

The spiral is another extremely common fractal in nature, found over a huge range of scales. Biological spirals are found in the plant and animal kingdoms, and non-living spirals are found in the turbulent swirling of fluids and in the pattern of star formation in galaxies.

All fractals are formed by simple repetition, and combining expansion and rotation is enough to generate the ubiquitous spiral.

A hurricane is a self-organizing spiral in the atmosphere, driven by the evaporation and condensation of sea water. Scale = 500 km = 5*105 m.

A spiral galaxy is the largest natural spiral comprising hundreds of billions of stars. Scale = 100,000 ly = ~1020 m.

The plant kingdom is full of spirals. An agave cactus forms its spiral by growing new pieces rotated by a fixed angle. Many other plants form spirals in this way, including sunflowers and pinecones etc. Scale=50cm.

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.

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.

I once saw an advertisement outside a bar for a happy hour special on yard of beer. This seemed like a great ploy: if the container is a yard-long straw, that’s not much beer. On the other hand, perhaps the container is a yard tall but has a square cross-section that is 7 feet by 7 feet, and you only get the great deal on the beer if you can finish all of it. Since that “yard of beer” exceeded the size of your body, you’d never be able to drink it.

First observation: A length cannot be used to specify a volume of beer.

Now suppose we have a pint of beer. How tall is it? By varying the size of the container you place it in, you can make it arbitrarily tall or short. How much area does the pint of beer take up when you spill it on the ground? It depends on how thick the puddle is.

Second observation: A volume of beer cannot be measured in terms of its length.

In general, physical objects in the world, like liquids, are 3-dimensional, and only have a 3-dimensional spatial measure. There is no meaningful way to measure them in 1 or 2 dimensions.

Many fractals have non-integral dimension, so it is not meaningful to speak of their (1-dimensional) length, (2-dimensional) area, or (3-dimensional) volume.

For example, the Cantor set is a fractal that might appear to have length, but does not. Here is the construction of the Cantor set:

1. Draw a line segment on the real axis from 0 to 1.
2. Erase the middle third of every existing line segment
3. Repeat step 2 forever.

How might one measure the length of the resulting fractal? You could stop it mid-construction and add the lengths of all the line segments — but that’s not the Cantor set. The final Cantor set, after step 3, has no continuous lengths, only points.

So, perhaps the Cantor set is 0-dimensional — then we could measure it by counting the points. Alas, it has an infinite number of points.

To define any meaningful measure a fractal, we must first determine its Hausdorff dimension. Here is the procedure.

1. Find some way to cover the fractal with n-dimensional spheres (line segments, discs, 3-d spheres, etc) such that if the radius of each n-sphere is raised to some power d, and these terms are summed, then the result is neither 0 nor infinity.
2. Write down the value d.

Here is a hand-wavy description of why the dimension of the Cantor set is log[math]_3[/math](2):

• Cover the section of the real number line from 0 to 1/3 with a line segment of length 1/3.
• Cover the section of the real number line from 2/3 to 7/9 with a line segment of length 1/9
• Cover the section of the real number line from 8/9 to 25/27 with a line segment of length 1/27
• Continue to cover the section of the real number line from 1-(1/3)^n to 1–2*(1/3)^{n+1} with a line segment of length (1/3)^{n+1} for n from 3 to infinity. Every point in the Cantor set is covered by such a segment.
• Raise the lengths of all these segments to the power log[math]_3[/math](2) and add them up
• The sum is finite, and is not 0.
• You just won’t find a way to get such a sum using a different covering and/or a different dimension.

Given this dimension, we might define a log[math]_3[/math](2)-dimensional measure for sets of the same dimensionality as the Cantor set, such as the Cantor set placed next a copy of the Cantor set that has been shifted 3 units to the right, or a Cantor set that began with a line segment on the real axis from 1 to 39.

It is worth mentioning that the method described above can be used to prove that the real line is has a Hausdorff dimension of 1, and that the Cartesian plane has dimension 2. I have a proof for this, but it does not fit within the scope of this question.

As an exercise, to try to contribute to the refinement of the subject, and to provide contrast, I'll try to make up my own answer from scratch.

Narrative fractals are the simple patterns found in any kind of story or sub-story. They can be sequenced in any number of ways. They can be found at any level of the story. Their expression and meaning turns out different when used at different points, at different levels, or in different contexts.

They are recognized at many levels. They make sense, they are felt emotionally and kinesthetically, there is a higher order to them, and they are expressed in the finer details.

Together, they make stories be experienced like stories. A coherence is formed and easily recognized by the experiencer.

Here are some examples:

* Dispersal & Concentration: It can be expressed with many different words. Divergence/Convergence. Distraction/Focus. It is two different patterns which are opposite of each other. In one, parts or actors are spread out in all directions, moving further away from each other and from the starting point, getting more divergent. In the other, the elements are moved into one place, or into a smaller place, getting closer to each other, more uniform.

One might imagine a wealth of stories and of patterns for different types of activities developed from just that. The agents in the story might be dispersed by some initial explosive event and sent on their respective adventures. Along the way they experience dispersals and convergences of various kinds, adding to their experience. Towards the end they'll all come together, the sub-plots all tie together and there's a coherent end for everybody. That's also, for example, how an Open Space conference typically develops.

* Attraction & Repulsion: Again, that's two different, opposite, patterns. There's something that attracts. The attractor might be a person or an idea or a place, or all sorts of other things. What is attracted might also be people, ideas, etc. Repulsion goes the other direction. Lots of stories are based on attraction and repulsion, which maybe shifts along the way. Maybe this pattern could be said to be exactly the same as the first one I mentioned, as Attraction will create a Convergence towards the point of attraction, and Repulsion creates a Divergence. So, they might possibly just be fractals of the same pattern. Except for that they kind of are guided from opposite directions. Attraction is a pull by something, which normally is known in advance. Whereas in a Concentration/Convergence the target is rather being constructed or found along the way. In Repulsion, there's something definite there that is repulsing, where as in Divergence, it rather seems like there's an external pull from many directions.

A story will often have some kind of obvious target, maybe the intent of one of the players. That intent would essentially be an attractor, but only for some of the people involved, maybe only for one person. That's maybe more linear? Does it need its own pattern? I don't know. Maybe at least:

* Flow. Inertia. Movement that continues in the direction it was going, and/or along the path of least resistance, like water. Until there's a…

* Challenge. Conflict. Barrier. There's something in the way, something one can't do, or something that blocks what was planned. Which might force some kind of…

* Transformation. Evolution. Change of levels, change of forms. Moving to a meta level, finding a new solution, or changing the game.

Since I mentioned Flow, I should notice that things aren't always in motion. We might also have a

* Standing Wave. An energy that can't flow.

and

* The Unexpressed. Potential energy. Maybe corresponding to consciousness. Or to anything latent, anything that feels like it needs to happen, but which hasn't manifested yet.

* Completion. Reaching a goal. Success and simultaneously Death, as the game is over.

This is a very quick, sloppy and incomplete collection of patterns that potentially might be candidates. Not something methodically thought through, tested and carefully selected. Simply to indicate other ways of looking at it. I'm trying to provide a bit of a challenge, not to propose a competing system.

I'd imagine a collection of patterns like those, which could be sequenced for different purposes, and which could be combined any which way, to create new deeper meanings. They should make sense in any order. Any two of them put together should have a clear meaning.

Of course for such a pattern language to be immediately useful, it has to be very simple to understand and one should be able to choose the appropriate pattern in a few seconds.

I'm thinking of de Bono's Six Thinking Hats for comparison. One can teach somebody to use them with a 2 minute explanation, they make sense right away, and one can usually categorize a statement as one of the six categories in a second or so.

Once one gets lost in "Hm, what might he possibly have meant by that?" or "Hm, that doesn't seem to fit here!" then the system breaks down. It wouldn't work as a system if there sometimes isn't any fit, or if one has to resort to lengthy explanations to make it fit.

Basket Starfish is a good example:

(from article http://www.telegraph.co.uk/earth/earthpicturegalleries/7875310/Scientists-discover-new-marine-species-in-the-hidden-depths-of-the-Atlantic-Ocean.html?image=1 )

And this is one of the less ludicrously branching ones!

It is a fractal that is topologically 2D. So it is a rough, undulating surface. Here’s a 2.5D fractal surface:

(Semi-dimensional shapes and other curves)

and here is a pyramidal fractal surface of varying bend angle:

(https://www.researchgate.net/publication/309391846_Three_Variable_Dimension_Surfaces)

In nature cumulus clouds have an approximately fractal surface:

as do rocks:

and in fact a huge number of natural surfaces are fractal over a certain scale and spatial range.

This is not surprising as locally fractal just means the surface area is locally a straight line (rather than a constant) on a log-log plot. And since smooth curves on a log-log plot are always approximated by straight lines over a small enough interval, it is really a self-truth that natural surfaces are fractal over some scale range.

Man-made surfaces are the pathalogical cases, being usually very smooth (Euclidean), often for manufacturing or cleaning purposes.

In popular culture, the Mandelbrot set has achieved immense fame. Many people who don’t know the name of it will have seen it in videos, pop art, t-shirts and so on. It seems like people got a little tired of it, but it’s still a famous images.

Inside of mathematics, the Cantor “middle third” set gets mentioned (relatively) a lot. I doubt there are very many mathematicians or mathematics students who know the Cantor set but not the Mandelbrot set. I suspect though there are more who know how precisely to define it, and can prove some of its basic properties. I suspect it appears in mathematics papers and textbooks more often too.

My impression is that this is partly due to its being a much simpler example. That the boundary of the Mandelbrot set has dimension 2 was considered a non-trivial research problem when it was finally demonstrated. Showing that the fractal dimension of the Cantor set is [math](\log 2)/(\log 3)[/math] is the kind of fact that could be assigned as an exercise for a class in which the fractal dimension had been defined. There are other examples of fractals where it is also not so hard to work out the fractal dimension (like the snowflake curve, which has dimension [math](\log 4)/(\log 3)[/math], but the Cantor set is maybe the easiest example.

The topology of the Cantor set (known as “Cantor space”) is just the topology of the product of a countable infinite family of sets each of which has just two elements. It’s used as an example of a variety of phenomena. It’s an infinite totally-disconnected space. (Given any two points [math]p[/math] and [math]q[/math], the space can be broken up into two disjoint open sets, one of them containing [math]p[/math] and the other containing [math]q[/math].) It has the same cardinality as the whole real line, but has measure zero (total length zero). If a metric space [math]X[/math] is non-empty and compact, then there is a continuous function from Cantor space which maps onto [math]X[/math].

There is a cute book called Counterexamples in Analysis by Gelbaum and Olmstead where a number of the examples are based in some way on the Cantor set. One variation on the Cantor set is known as a “fat” Cantor set. The Cantor set has measure zero, but if instead of removing the middle third at each step in its construction, we remove smaller and smaller portions, it’s possible to get a final result which is sort of like the Cantor set but which has a positive measure. A fat Cantor set is an example of a set with a positive length, but which doesn’t include any interval of positive length. It’s nowhere dense, meaning if you pick a point in the set and an interval of positive length around the point, there exists a sub-interval of that interval also of positive length which doesn’t contain any of the points of the set.

I can’t think of any other fractal that is mentioned as often inside of mathematics.

The Mandelbrot set has some mathematical interest to it, but it isn’t mentioned as often. It is just somewhat more esoteric for a mathematician than the Cantor set. It would be much easier to go through a whole mathematical education and not run into it in any classes or papers. Before I went to graduate school, the only way I’d seen it mentioned was in popular math articles, books, and documentaries. I seem to remember that there was an episode of the PBS series Nova in which Mendelbrot himself was interviewed and where the film makers said something about his well-known set. In graduate school, one of my fellow graduate students put together a seminar talk in which she proved various of its standard properties.

Technically, the Mandelbrot set is not a fractal; it is a two-dimensional closed set in the complex plane. The fractal dimension (also known as “Hausdorff dimension”) of the boundary of the Mandelbrot set is also 2, however, and it counts as a fractal. The dimension isn’t fractional (so the boundary is not literally “fractional dimensional”) but generally when the boundary of a region in the plane has fractal dimension greater than 1, one considers it to be a fractal. The fact that the fractal dimension is greater than 1 shows that it’s gnarly in a way (and dimension 2 shows it is a little surprisingly gnarly).

Also, I don’t know of anybody who is seriously bothered by the Mandelbrot set itself being called a “fractal”, even if it doesn’t fit the definitions of “fractal” that I’ve heard. I remember one professor who would tell graduate students to think of “fractal” as meaning “closed set”. Taking this definition literally wouldn’t be good for answering a question like this one. I would consider it silly to say to you that circles and lines were more famous examples of “fractals”. But the concept of “fractal” is also not one where having a precise definition pays off to any meaningful degree. Even mathematicians who are studying sets that one would consider to be “fractals” can ignore these attempted definitions of the word and it really doesn’t matter.

Fractals is what dreams are made of. It is like finding system in chaos.

In the simplest terms it is a complex geometric shape that is constructed using repetitive equations. There are fractals which have infinite perimeter but finite area, then there are fractals which are bound figures technically having no area . One of the most interesting thing about fractals is that if you take a small part of the given fractal, it would look exactly like the whole.

Much like the Fibonacci sequence or the Golden Ratio, they are everywhere in nature; starting from clouds to shell. Seems like God knew more maths than anybody.

Here are a few gorgeous fractals (They super-models of the maths world!):

Ok, but don’t blame me for the answer, u asked for it :D. I have actually wanted to post something mindbending for my question - but I saw this question so… It can wait. Maybe there is someone Curious sitting behind this question :)? Nothing happens without a cause, there are no coincidences - there are only co-incidences.

Let’s get into it then. We will make a quick experiment.

Sit back comfortably and think:

1. You are here in your head, your mind, between your ears. - so it is YOU.
2. You are here as You - your body. - so it is PHYSICAL YOU.
3. Back to head. You can look in your mind next to those two guys above. - so it is THIRD YOU. An Observer.

Now, you can do same thing through Third’s eyes and you see those 3 above. Step next to it again and you will see 4 “YOUS”, etc.

So let me clear it up:
1.

2.

3.

4.

5.^N

Now what is really funny - the fractal starts after first 3 “YOUS”.

It’s like Pi - 3 and then “,” and up you go to N. So well - there you have a fractal in mind. Or you can even say it’s a visually logical fractal. How funny it’s so similar to Pi institution… ;)

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?

One of the properties of a fractal is that they look similar at any scale.

Coastlines are an example.

Or think about this picture why did the photographer place the geology hammer in the photo. Because without it, you could not be sure of the scale of the rocks, it could also be a microscope image of gravel. Many things in nature look similar at different scales.

I nominate Power Tower Fractals as needing the most challenging math.

A fractal is a mathematical idealization that does not exactly apply to any real-world objects, because at a small enough scale they (and we) are made up of atoms, but there are some features of many living creatures that resemble fractals.

Given a (bounded) set of points in space, one can define a covering function [math]c(r)[/math] which given a radius [math]r>0[/math] equals the number of balls of radius [math]r[/math] (possibly overlapping) that are needed to cover the set (contain it). A ball is like a sphere but solid (including the inside of the sphere). The rate of growth of this function being “anomalously” high is what makes a set a fractal. For example, the “snowflake curve” is a curve where [math]c(r)[/math] grows like [math](1/r)^{(\log 4/\log 3)}[/math]. When I say “grows like”, I mean that it stays within a constant factor of that trend. The exact value grows a bit irregularly as [math]r[/math] goes to [math]0[/math]. But that it grows faster than [math]1/r[/math] shows that the snowflake curve is unlike a non-fractal curve. Dividing [math]r[/math] by [math]3[/math] causes [math]c(r)[/math] to increase by a factor of [math]4[/math], not [math]3[/math]. The snowflake curve is self-similar, but a fractal does not need to be self-similar.

Human beings (and most multi-cellular living creatures) have some organs that are like fractals. Our lungs have big branches that fork off into smaller ones, which fork off into still smaller ones, and so on. That makes the surface something like a fractal. The shape is not exactly a fractal; once you look closely enough at the smallest branches, you come to an end of the branching. If we define a covering function [math]c(r)[/math] for the surface of a lung, it will stop behaving the same way (growing on the same trend) by the point [math]r[/math] gets smaller than a single lung cell. If [math]r[/math] is smaller than the diameter of a single atom, defining which portion of the atom is “on the surface of the lung” becomes a strange (and somewhat meaningless) question. I would be inclined to count some atoms as being completely part of “the surface of the lung”, which makes the surface a three-dimensional solid. However one defined it as a set of points in space, though, it won’t be a fractal.

All that is a bit of a nit-pick though. When people give “real-world” examples of fractals, they are all shapes that look like fractals on some range of length scales. A sea-shore resembles a fractal curve, although that breaks down at a small scale. A roadway system is a bit like a fractal, although it clearly breaks down at the scale of the individual vehicle. Lungs, going from the overall size of the lung down to its tiniest branches, are very much like fractals in the same way. The blood vessels also have a hierarchy of different sizes that makes the structure of their walls like a fractal. Eventually one gets down to capillaries (which let one blood cell through at a time) and the scheme breaks down, but that’s okay.

This is my favorite one, Dragon Curve. I like Dragons. They are big and if someone tries to mess with 'em they burn them. But here:

Take a strip of paper, A VERY LONG strip of paper! (although it is impractical, just think you did get one)

Fold it once(end to end) and then unfold it, look at how it aligns itself, the vertex is a fold: (here is the side view)

Let's do the same one more time:

yet again:

and, again:

once more:

take a break. this is getting hard. Let's do it one more time:

Woo! 6 folds, that is [math]2^6[/math] layers of paper. I think we can do one more:

Now, Imagine (we can't do any more folds, oh wait, this cannot be imagined, here is what computer does):
after one more fold:

starting to look like a dragon? Pretty Much. another one:

Ooh, taking a shape. Let's do 1 more fold:

Ahoy! 1 more:

Another one captain` Aye Aye!:

Keep going:

I said, keep going:

Wooh! This is what it will look like after infinite folds:

Like a dragon!

There is more math to this curve.

[math]l=sqrt{2}*d[/math]and so on for other such segments for the curve.

EDIT: A fun fact. if [math]a[/math] was the original length of the paper then the area of the curve would be [math]{a^2}/{2}[/math] after infinite number of folds. The mathematics behind that is beautiful. I am too lazy to type that out. :P

Well, fractals themselves aren't chaotic. The mathematical way of stating (one of the) link(s) between chaos theory and fractals is "basins of attraction with fractal boundaries exhibit chaotic behaviour". I'll explain what that means with an example.

Imagine that we have a physical system of the following form. There is an object moving through two-dimensional space, a marble rolling, with position [math]s=(x, y)[/math] and velocity [math]v=(v_x, v_y)[/math]. Imagine that we have some physics equations that let us calculate how the marble's velocity and position changes with respect to time. We can see the states of this system as points in four-dimensional space: [math]z=(x, y, v_x, v_y)[/math]; we can see the evolution of this system with time as a trajectory through a four-dimensional "state space".

Now let's say that this system has two possible equilibria, two stable states, that it could tend to. For example, imagine that this system was a rolling marble on a bumpy surface with two dips in it, such that the marble is going to fall into one of those dips and stop moving. We want to calculate into which hole the marble will settle and stop moving.

Now further imagine that, through mathematical methods not necessary to mention here, we've calculated exact results for which combinations of positions and velocities will, guaranteed, lead to the marble settling into which hole. Formally, imagine that we have a subset D of our four-dimensional state space, a group of positions and velocities, such that every point z in D falls into hole 1, while every point z that is not in D falls into hole 2. That is, if the marble ever enters the region D, it is guaranteed to fall into hole 1.

We call D the basin of hole 1. Let d be the dimension of the basin D. For example, if the space D is defined as "if x is ever less than 1, the marble falls into hole 1", then d is 1. If D is instead defined as "if ever x is less than 1, [math]v_x[/math] is less than 1, and [math]v_y[/math] is less than 2, the marble falls into hole 2", then d is 3.

In the real world, we don't have exact measurements - we don't observe the position and our velocity of our marble directly. We observe that the marble is at a position [math]\hat{z}[/math], although our sensor isn't perfect. Our sensor does tell us, though, that the true position z is within a distance r of [math]\hat{z}[/math]. So in our four-dimensional space, the possible true positions z form a hypersphere, centred on [math]\hat{z}[/math] and with radius r.

Now is where it gets interesting. Remember, we want to calculate whether the marble, if we observe the position [math]\hat{z}[/math], is likely to fall into hole 1 or 2. If we assume that [math]\hat{z}[/math] is right on the edge of D, this task is difficult. We want to calculate what fraction of the hypersphere of possible true positions is inside the space D, versus what fraction of the true positions aren't.

Now, the formula for the volume of a hypersphere with radius r in n-dimensional space is complicated, but all that we need to know is that it is proportional to the radius to the power of the dimensionality of the space: [math]V(n)=Kr^n[/math]. And so the fraction of true positions that are in the basin D for hole 1, to possible true positions in the four-dimensional state space, is [math]f=V(4)/V(d)=Cr^{4-d}[/math].

If there is no chaos theory at play, and our measurements are good enough, this problem is solved. For example, assume the dimension d of the basin space is 2. Then, the fraction is [math]f=Cr^{4-2}=Cr^2[/math]. And therefore, as we decrease the radius of our error, the fraction of a given observation in the basin space decreases quadratically. This means that as our measurements get more accurate, we get much more information about whether or not the marble will end up in hole 1. For example, imagine that we make one measurement with error [math]r=1[/math], and another with error [math]r=1/4[/math]. The first error gives a fraction proportional to [math]f=C[/math], while the second gives [math]f=C/16[/math]. Increasing our accuracy by a factor of four increased the accuracy of the estimate of the correct equilibrium by sixteen.

However, sometimes basin edges can have fractal dimension d. That is, the dimension d can be a fraction. And if that fraction is close to the dimension of the state space, which is in our case 4, the problem can become much more difficult. For example, imagine that [math]d=3.9[/math]; that gives [math]f=Cr^{4-3.9}=Cr^{1/10}[/math]. Now, increasing the measurement accuracy - decreasing r - doesn't help us tell whether the marble will fall into one hole or another. For example, imagine that we increase the accuracy of our measurement by 100 times: [math]r_2=\dfrac{r_1}{100}[/math]. That has much less effect than before: [math]f_2/f_1=\dfrac{r_2^{1/10}}{r_1^{1/10}}=\dfrac{r_1^{1/10}}{100^{1/10}r_1^{1/10}}=100^{-1/10}\approx 0.631[/math]. To increase the estimate accuracy by a factor of sixteen, as we did before, would require increasing the measurement accuracy by a factor of over one trillion (!!!)

So the bottom line: when a basin of attraction D's boundary has fractal dimension, it can be very hard to tell whether a given noisy observation [math]\hat{z}[/math] represents a point that is going to converge to the equilibrium in D, or not. This is not the only interaction between chaos theory and fractals, but to me it's the most straightforward.

Fractals a a nice way of storing information, particularly when you are describing the behavior of a group of objects that all act similarly to each other. It is related to the idea of recursion.

A tree might look like one object, but it is many little cells working together, each cell operating off of the effects its environment has on it and the instructions in its DNA. As the tree grows, each cell acts on different parts of the instructions of its DNA and performs different actions, resulting in branches growing some places, roots in others, and leaves growing at the top.

The geometric figures we study in high school, such as triangles, rectangles, and squares, do not really occur in the real world, but they are nice simplifications of structures that do exist in the real world. Similarly, we cannot usually work with the whole structure of a tree, or a brain, or any of these highly complicated structures, but by looking at the individual units of the brain, or the tree, or whatever, we can come up with simplified sets of instructions, and then use fractals to create models of the whole structure.

This is how many of the background design in videogames and CGI movie scenes are performed. The artists cannot color everything to the detail needed, so they create a short list of rules and have the computer execute the code. This way, when you are far from the mountain range, the computer doesn't need to do all of the detail of the mountain. They can just run the first few steps of the fractal. But as you walk closer, the computer starts drawing more of the fractal, giving more detail as it is needed.