This list illustrates most of the fractals known to me:
List of fractals by Hausdorff dimension

The fractals I consider to be the most interesting are obtained from solutions to mathematical models/approximation of physical systems.

(Feigenbaum attractor, found in logistic map type systems like the Lotka-Volterra system)

(The Ikeda attractor)

(Ueda's attractor)

(Henon attractor)

(The Rossler system, you find this in competitive Lotka-Volterra type systems)

(Of course, the Lorenz system)

(The dendritic Julia - Neural communication networks/lightening/fracture models)

(Weiner process/Brownian motion/Drunkard's walk. Very useful in introducing stochasticity in models)

The L-system branching

(A percolation front at threshold)

(3D and 2D diffusion limited aggregate objects, and a Leichtenberg figure example of a DLA )

(Brownian surfaces)

Because I believe self-similar is an efficient way organise structure in an finite plane or space. Take the Koch Curve for example, it can have infinite length in a bounded area.

We can also find this phenomena from sea shells and spiral galaxies to the structure of human lungs, the patterns of chaos are all around us.

For plants example, more space means more photosynthesis, which means more chance to live on.

Forget fractals for a minute. Instead let's make a simple game called "Microscope."

We start the game looking at one character in our microscope:

x

(This x doesn't represent anything! It's just an x. Like in Tic-tac-toe.)

In Microscope, any time we want to see more detail, we can zoom in.

Rule: to zoom in, replace all x by xox and replace all o by ooo

Now let's zoom in:
x --zoom--> xox
xox --zoom--> xoxoooxox

Each time we zoom in we are magnifying the picture 3 times. (Right? It looks 3 times as wide as before.)

So an x seen at 9x magnification looks like this: xoxoooxox. You can zoom in as far as you like.

Try it yourself.

Or we can pick part of the view and just zoom in on that. "A universe in a grain of sand," right?

So what does self-similar mean? Self-similar means if we pick any part of the fractal (the x's are the fractal) and zoom in on that, it looks like the whole thing.

For example take xoxoooxox and throw out everything except the leftmost character. We have:

x.

Now forget the rest of the pattern. zoom in on just that x. We'll get the entire fractal inside just that little bit, same as if we kept zooming in on the whole thing.

That's an example of "self similarity." Every little bit looks exactly like the whole. Not all fractals have this property, but they can have it.

I now grant you the authority to make up games like this yourself. Forget about reading what people say about fractals for a while. Just screw around with little games like this.

If you decide you want to read something, my favorite book is Fractals, Chaos, Power Laws by Manfred Schroeder: http://www.strandbooks.com/physics/fractals-chaos-power-laws-0486472043/_/searchString/fractals%20chaos%20power%20laws
You don't have to read his chapters or sections in order. Just pick around in it.

Well, you could say reality is self-similar, but the notion that there is any one thing that exists in isolation is erroneous and so looking for an independent thing which has the alleged traits of self-similarity is therefore a fools errand because there simply is no object in isolation…

Basically reality is inseperable from itself. It may appear fragmented and filled with independent objects to us, but that’s a function of perception. It is not a fundamental quality that exists in some hypothetical seperate object.

It’s simpler and more accurate to simply say that whatever you happen to be looking at and despite the apparent differences between things, you are nonetheless always observing the same thing (reality).

Paradoxically though, because things appear to us in the manner they do as a function of perception, this means there is no real “thing in itself” that we could ever know even if there were truly independent objects that existed because how things appear to us (and indeed anything else too) is dependent on the particular idiosyncrasies of the perceptual system engaged in the observing process.

In other words, how things appear is determined by the structure of the perception system used to observe…If I had eyes which couldn’t process colour, things would appear to lack colour, but who’s correct in their view of what colour the thing really is?

The answer is everyone and no one. Why? Because colour (and indeed lack of colour) are things that are seen based on the capabilities of our perceptual systems. They are not intrinsic to the object itself.

Examples of fractals in nature

The natural fractals only exist between certain scales, that are unique for each type of fractal.

Fractals are often seemingly created from very simple rules, but the rules may differ between different examples of natural fractals. It’s often not possible to know exactly why the fractals appear as they do, or exactly what underlying rules that are in effect. We only see the end result. The Romanesco broccoli below is generated by genetics.

Ice crystals are generated by physics rules in combination with environment.

Ferns are created by genetics.

Tree branches are created by rules from genetics, in combination with environment.

River flows are created by physics and environment.

The cosmic web is a result of physics (gravity and dark matter and dark energy).

Footnotes

The Giants Causeway Northern Ireland is made up of five sided Basalt formations.

It doesn't get much more obviously self-similar than a Romanesco Broccoli. When someone asks me what a fractal is, I usually point out one of these as the first example:

Mathematical examples are given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve,space-filling curve, and Koch curve.
Chaotic dynamical systems are sometimes associated with fractals.Objects in the phase space of a dynamical system can be fractals (see attractor).

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:
Exact self-similarity
Quasi-self-similarity
Statistical self-similarity

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, river networks, fault lines, mountain ranges, craters, snow flakes,crystals,lightning, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels, and ocean waves. DNA and heartbeat can be analyzed as fractals. Even coastlines may be loosely considered fractal in nature.

The statements above are from http://en.wikipedia.org/wiki/Fractal

Your can follow the link for more detailed examples. As I know, computer generated landscape such as cloud ,mountains, rivers in many modern movies are constructed using fractal, this perhaps is the artificial fractals people see most frequently. Also, the fractal analysis is used in many area, including fraud detection in data mining, fault diagnosis in industrial engineering , fractal image compression, etc.

Originally fractal geometry referred to recursive shapes like the Koch curve, which are all self-similar by the very fact that they are recursively defined.

Later fractal geometry was generalised to any shape with Hausdorff dimension greater than its topological dimension. This includes stochastic shapes which are only self-similar in a statistical sense.

But you could make a curve that looks like a Koch curve at one scale, and a Minkowski sausage closer in, and a Dragon curve as you zoom further, then a Levy C curve at greater zoom, etc etc. You could even parameterise the style of the curve based on the digits of pi so the look never repeats. But yet they can all have the same Hausdorff dimension, so it is an example of simple (mono) fractal geometry… but it is not self-similar.

So self-similarity is just a rough description that is helpful for common fractals. It isn’t a rigourous definition.

It means the thing looks similar at many different scales. In its most strict it means a shape that is the same shape after some scaling (dilation). The phrase ‘same shape’ here means that you can apply Euclidean transformations (rotation and translation) when matching.

Their importance comes down to history…

The Greeks built up the idea of shapes (and in fact geometry in general) around symmetries, specifically translation, reflection and rotation symmetries. If you look at objects with these symmetries you will find a list of shapes with very Greek names, such as the Platonic solids (octahedron, dodecahedron etc) after Plato, the Archimedean solids (after Archimedes), and in general Euclidean geometry (after Euclid) which consists of smooth lines and faces.

For 2000 years geometry has been based on this, but the ancient Greeks left out the remaining symmetry: scale. If you add in this symmetry then you get objects like spirals, and most generally (a bit of all the symmetries) you get fractal structures. This is now called fractal geometry, and has only really been around since the 1960s, since it took the invention of the computer to really be able to work well with these structures.

The interesting thing about including this last symmetry is that the structures you get are much closer to natural ones than the harsh and artificial lines and planes of Euclidean geometry. Instead you get structures like lightning, rough rocks, crumpled surfaces, cracks and mountainous looking terrain. In short it better represents what we see all around us.

So no longer is geometry about maths being a pure and artificial ‘perfection’ of the crude shapes of nature, instead it is a language that describes nature and sees how well it works. As Mandelbrot (who pioneered fractal geometry) famously said: clouds are not spheres, mountains are not cones and lightning doesn’t travel in straight lines.

So, in short, most things in nature that have symmetries have some scale among those symmetries. This produces all the rough and complex designs that we see around us. Rocks, mountains, cracks, trees, rivers, etc etc. That’s why the topic is important, it gets back to the ‘geo’ part of geometry, which is about measuring the real world (geo), not just smooth and straight abstractions like polygons.

What are “things”? If they include molecules, then the assertion is false, since nearly all oxygen molecules are exactly alike. Nearly all carbon dioxide molecules are alike. Nitrogen. Water. Methane. Also atoms of noble gases like neon, argon, … Etc. So the air is full of exactly alike things. The oceans are full of exactly alike things. Etc.

The only differences among a particular type of molecule would be different isotopes of an element.

Even if you restrict “things” to refer only to what humans make, if you accept “parts of things”, then with an atomic force microscope you can create a simple pattern of atoms that is exactly replicable. However, the pattern would be placed on a substrate, and no two substrates would be exactly alike.

Evolution is a divergent process. This means that, all things being equal, lineages will tend to become increasingly different from one another over time. Once two populations stop sharing genes the random mutations that occur in one will not spread to the other, and since those mutations are random, they will be different. And if the two populations end up in different environments, then the divergence will be accelerated by different selection for different traits.

Similarities between organisms arise by two main mechanisms. The first is homology, ie common descent. The more closely related two organisms are the more likely they are to look similar, or have body parts that look similar.

The second is convergence. Wherever shape has functional implications, certain shapes will be advantageous for certain circumstances. If two lineages live in the similar environments and are subject to similar selection pressures on appearance, they will both over time be selected to converge on that optimal shape and appearance.

One of the things I really enjoy about macro photography is seeing the symmetry of nature. Here is the image of a succulent plant in my garden from yesterday. Truly amazing.

Two trees (and for that sake all organisms) are not exactly alike in a narrow sense, but they have broad similarities— both structural and functional. The reason:

“Biological systems are not mathematical systems.” In biological sustems 2=2 is not a uniform 4, but it is 3.8 or 3.999 or 4.001 etc in biological systems. This is because the end results are multifactorial.

The fifth leaf of a mango plant shall never be the same as the fifth leaf of another mango plant, but it shall be the mango leaf and not the peach leaf. Apply this to your siblings, twins, and else.

Have your blood sugar measured 3 hours after eating 50-gram wheat flour on two successive days at 10 am. You shall get a very similar range of values but not the same.

What you see in the tree (as above in the question asked) is its form— phenotype. Form (phenotype is a complex interaction of intrinsic program (genotype) and environmental factors.

I'm not sure this is precisely relevant to your question, but I can think of two interesting systems you may want to take a look at.

In 1987 I spoke with Stephen Wolfram about the possibility of physics being based on iterated functions. Wolfram opened a copy of Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid and showed me a fractal picture formed from the distribution of electrons in a magnetic field. The fractal of course expresses a self-similar pattern repeating at different scales. While I do not know for sure whether this pattern propagates from a smaller scale to a larger one or the reverse, I suspect that in physics these sort of patterns tend to propagate from larger scales down to smaller one.

In the phenomena of turbulence eddies do begin at a larger scale and then propagate downward into smaller scales in what I understand to be a self-similar manner. This occurs until a small enough scale is reached that the fluid transforms all the mechanical energy into heat.

The shape of crystals directly reflect a pattern of organization at the molecular level. If the atoms form a cube, you'll get big crystals that look like cubes.
You could also include snow flakes as an example.

No two things can be exactly the same because they can’t both be in the same place at the same time. There are certain degrees of similarity for things that are in different places and affected or determined by their situation. If we restrict ourselves to objects in the everyday world that we perceive with our senses, then we can understand why we have the concept of “similarity” or sameness or equivalence. We have phrases such as “like as two peas in a pod.” Not everybody looks into a pod to check this. We could say like as two peas in a can or a frozen package. If it mattered, we would always be able to set them apart. It just doesn’t matter.

So what determines how much differences matter or not? If I write a message with a pen on paper, my letter “a” won’t be like your letter “a” but as long as it’s functionally recognizable as a letter in a text, then it’s the same as all other recognizable letters. And if I say the word “same” in my voice with my accent, it’s going to be a different sound from your voice and your accent, but as long as we recognize the word, then it’s the same word. The differences belong to our identity, not to the word.

Martin Heidegger (pbuh) gets into this a little with his attempt to find the essence of truth in an essay first published in 1943. He takes the example of two five mark coins. As money circulates, it gets battered and worn. In manufacturing a metal disk, the material will surely reveal small differences if we test it accurately enough, but they do not signify as long as the coins are both “real” and our request to buy something with them must be honored because they possess the value of five marks for a monetary exchange. That is the test of truth in their case. All such coins are “true” bearers of their value as long as their provenance is believed in. They are all the same.

He says the statement or claim of their truth is only a sentence, and as language, it is absolutely different from the substance of the objects themselves. A coin is round, but the statement about them has no spatial character. Nor of course would a sentence be taken as legitimate exchange in a purchase. Although one can question that because money is actually a form of promise — read the text on a dollar bill. Moreover, when two people make a deal and shake hands, it can have the same value in the force of the promise as the presence and transfer of currency.

Heidegger then goes on to discuss the idea of “true” gold as opposed to fake gold. He wants to keep the principle, already open to question, about statements and objects. But the golden appearance of an object only becomes “untrue” in the context of a statement about it. It’s only fake or untrue if I say it’s real gold. In itself, it’s simply what it is. If it’s iron pyrite, then it’s true iron pyrite as long as no one tries to pass it off as gold. If one is looking for a chemical that combines iron and sulphur, gold isn’t the true object of your desire. Of course you could sell the gold and buy yourself some iron or some sulphur, but that would depend on someone else believing what said to them about gold, which ultimately depends on gold as a symbol of value in a community of exchange.

Everything is what it is, and everything we say is what we say. Similarity depends on the world of claims and meanings and symbols.

In 1943, Heidegger was aware that his attempt to find a similarity between his view of true Germanness and the National Socialist view were not the same. He was also aware that the discrepancies between what Adolf Hitler was declaring about the war and the power of the armies he was confronting was not the same as the actual forces at work against him. The truth in the realm of weapons and destruction lay with the defeats of El Alamein and Stalingrad. That was not like the picture of a world reshaped by German power and manifesting a German destiny.

People used to tell me about Donald Trump that he “tells it like it is.” The world that he depicts in his words is the real world because the picture in his words is the same in that correspondence to the world we will see when we look with the right convictions. Do we all live in the same world? No, we do not. We are all within a different situation, a different moment defined by who we are and what ends we are pursuing. And yet, like the letter “a” we are also all alike in the value and the meaning of our lives. Those with power are still just like those without power. Those with knowledge are just like those without knowledge. But the power is not alike and the knowledge is not alike. Then it comes down to another question, just like the peas in the pod. What matters?

We have an easy example in domestic dogs and wild wolves.

We can easily make domestic dogs look quite different to their wolf ancestors, just by artificial selection.

But we can also easily ensure that domestic dogs look about the same as wolves, again by artificial selection.

It is all about natural selection if we are not doing the selecting, and as Charles Darwin explains the basic principles quite well in his 150 year old book, I could even ask that you read it, as long as you understand that it is 150 year old science.

Modern science can tell you how it works on a “molecular level” too, because we know about DNA, that Darwin did not know about.

The answer is just natural selection.

For example, our common ancestor with chimpanzees would have looked more like a chimpanzee than a human.

The Australopithecus fossil we discovered called Lucy clearly was a possible human ancestor, because she walked upright, but she had a brain the size of a chimpanzee.

If we are related to her immediate family (from what I understand she had not had any children), then it is clear that we evolved larger brains after we evolved walking erect.

So a good way to speculate what was the driving force in our separate evolution to chimpanzees is that she walked upright on the open savannah, where there are no trees.

Our chimpanzee cousins do not do that. They are just as vulnerable to the big cats as us - if not more so.

• We are a more co-operative species than chimpanzees.
• If we have sticks and stones as “tools” we know how to use them better than our chimp relatives.
• If we have time, we can make those tools even better, and unlike chimps, keep them with us for use as needed.

Our appearance is best explained as the CHOICES our ancestors made.

In this case, coming out of the forest, carrying our fabricated tools with us, and making it so obvious that if we are in a group, and the lion can see the whites of our eyes, it will be the lion that dies in a confrontation.

Besides, the lion will already know that one of us will not be that much to eat. There is other game it can pick on with a lesser chance of dying.

Our appearance is the result of evolution, of which the “information” comes only from the natural selection.

One of the classic examples comes from Odum's classic text “Fundamentals of Ecology. The predictor- pray relationship between lynx and hare as evidenced by ten years of pelt data collected by the Hudson Bay Company.

Another example is the relationship between population size and carrying capacity which is understood as a oscillation; a limit cycle.

The effect on the environment is equilibrium; however there is often hysterisis ( a delay) in the response in natural systems where equilibrium is biologically mediated.

Some, including a few economists, refer to hysterisis in natural and human systems as a “J" curve phenomenon.

They don't. What does and this is the key feature of self similarity is that they are simple to program. Self similarity means that X=union f_i(X) where f_i(X) are functions that merely translate rotate or shrink the original set. Fractals are any geometric figure that has a non integer dimension. 3B1B and a discrete textbook whose name eludes me explain this well. So the key way we are defining dimension is that if you scale the length of the fractal by s, the mass of the fractal grows by a factor of s^d. Self similarity is often used because it's easy to demonstrate the scaling by having a copy of the original and self similar fractals are generally considered more aesthetically pleasing than the coastline of Britain or Norway which are also fractal with dimensions around 1.2 . In the case of the sierpinski triangle halving the base gives you three copies of the original fractal so the dimension should be 2^d=3 or log(3)/log(2). For the Von Koch Curve cutting it in a 1/3 length gives you 4 copies so you get 1/3^d=1/4 or 3^d=4 or d=log(4)/log(3) or d=2log(2)/log(3). A square is also self similar with dimension 2 so not a fractal. For more general fractal counting boxes as you make the boxes smaller plotting on a logarithmic scale and finding the slope is how the dimension is calculated.

Self-similarity is very common... two that come to mind:

• Tree branches. http://harryseldon.thinkosphere.com/files/fractals_nature_model.jpg
  
• The archetypal plant fractal, a fern.
• More: http://www.google.ca/search?q=fractal+beauty+nature

Though I am not a biologist or scientist, it seems to me that the same mechanisms are responsible, while scale is a "parameter", by perfect analogy to mathematical Recursion. Another example is host/parasite relationships. It is not that the existence of a small scale parasite triggers a larger one, but that the "pattern" of parasitism turns out to be viable at any scale. http://c2.com/cgi/wiki?FleasAdInfinitum

Whole books have been written on the subject:

• http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869
• http://www.wolframscience.com/

The partitions of an integer are self similar. Also, the partitions of n+1 are "similar" to the partitions of n. This isn't a surprising property of Recursively defined functions; self similarity is a visual signature of recursion.

12  
11 1  
10 2  
10 1 1  
9 3  
9 2 1  
9 1 1 1  
8 4  
8 3 1  
8 2 2  
8 2 1 1  
8 1 1 1 1  
7 5  
7 4 1  
7 3 2  
7 3 1 1  
7 2 2 1  
7 2 1 1 1  
7 1 1 1 1 1  
6 6  
6 5 1  
6 4 2  
6 4 1 1  
6 3 3  
6 3 2 1  
6 3 1 1 1  
6 2 2 2  
6 2 2 1 1  
6 2 1 1 1 1  
6 1 1 1 1 1 1  
5 5 2  
5 5 1 1  
5 4 3  
5 4 2 1  
5 4 1 1 1  
5 3 3 1  
5 3 2 2  
5 3 2 1 1  
5 3 1 1 1 1  
5 2 2 2 1  
5 2 2 1 1 1  
5 2 1 1 1 1 1  
5 1 1 1 1 1 1 1  
4 4 4  
4 4 3 1  
4 4 2 2  
4 4 2 1 1  
4 4 1 1 1 1  
4 3 3 2  
4 3 3 1 1  
4 3 2 2 1  
4 3 2 1 1 1  
4 3 1 1 1 1 1  
4 2 2 2 2  
4 2 2 2 1 1  
4 2 2 1 1 1 1  
4 2 1 1 1 1 1 1  
4 1 1 1 1 1 1 1 1  
3 3 3 3  
3 3 3 2 1  
3 3 3 1 1 1  
3 3 2 2 2  
3 3 2 2 1 1  
3 3 2 1 1 1 1  
3 3 1 1 1 1 1 1  
3 2 2 2 2 1  
3 2 2 2 1 1 1  
3 2 2 1 1 1 1 1  
3 2 1 1 1 1 1 1 1  
3 1 1 1 1 1 1 1 1 1  
2 2 2 2 2 2  
2 2 2 2 2 1 1  
2 2 2 2 1 1 1 1  
2 2 2 1 1 1 1 1 1  
2 2 1 1 1 1 1 1 1 1  
2 1 1 1 1 1 1 1 1 1 1  
1 1 1 1 1 1 1 1 1 1 1 1