股骨头坏死起床腿酸疼:分形几何概述

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FRACTALOGY 101

By Khoa Nguy?n - Nov 17th, 2008

Tag: article, fractal

Are you familiar with fractals in general? Have you had your daily share of fractals yet? If you have gone out of your house today, I suggest that the answer is yes. It does not matter if you answer no since I’m pretty sure that you have seen more than one fractal before. Maybe you haven’t noticed but fractals are all around us. They play an important part of nature, and even our universe if I might say so.

Are you familiar with fractals in general? Have you had your daily share of fractals yet? If you have gone out of your house today, I suggest that the answer is yes. It does not matter if you answer no since I’m pretty sure that you have seen more than one fractal before. Maybe you haven’t noticed but fractals are all around us. They play an important part of nature, and even our universe if I might say so. Either ways, here’s an article written for your pleasure to get to know fractals better. For starters, we will look at some examples of that strange concept:


  

 


At least these are not too strange, I hope :D They are things we see in our daily life, and they contain elements of fractals. Clouds, ferns, soil patterns, etc. can be considered approximate fractals in nature.

 

So what are the similarities among those I have mentioned? At first glance, the images above all have somewhat irregular branches. Maybe that is a property of fractals? You have figured one out, congrats, but there are more left. Well, if you do some exercise with this fractal thing, you’ll find yourself in a certain website whose name begins with ‘W’ and ends with ‘ikipedia’. This certain website has an article with full descriptions of fractals, and you can go there to read about this. But I will still list it here for your convenience.


Characteristics of fractals

 

Let’s see what Wikipedia :D has in store for us:


 


A fractal often has the following features:

  • It has a fine structure at arbitrarily small scales.
  • It is too irregular to be easily described in traditional Euclidean geometric language.
  • It is self-similar (at least approximately or stochastically).
  • It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
  • It has a simple and recursive definition.
 


Just skip any lines that make your head spin. We’re not talking about complicated things today. So, a fractal is something recursive, something that has similar parts which are replicas of the whole object, something that repeats itself simultaneously. In other words, fractal is a fractal is a fractal is a fractal – can there be anything simpler? Fret not if you don't get it, the animation below should explain everything.




With these properties, even the broccoli and cauliflowers we avoid eating every meal are fractals, here’s the horrendous proof:


 

One branch of the cauliflower has about 13 smaller branches and in one smaller branch there are about 13 smaller-er branches, no wonder they taste weird ( ̄▽ ̄ ) The power of nature is just that terrifying. So can we reproduce these results? In computer graphics, fractals can be created using many methods. As this is an introductory article, we will go through the simplest one.


Construction

 

Imagine that we’re designing a program which draws lines when we input some command. Let 0o be the initial angle between the program’s ‘pencil’ and the horizontal. The following elementary commands are given:

 

F: “draw a line segment with determined width and length”.

+: “turn left an angle α”.

-: “turn right an angle α”.


We choose α = 90o for demonstration. The combination F+F-F-F+F should give us this graph:




To make things complicated, we give a rule like this: F->F+F-F. This means that the program must replace every F in the initial combination with F+F-F. For example, from a single F, the first iteration gives us F+F-F. After two iterations, the result is F+F-F+F+F-F-F+F-F. After three iterations, it becomes so lengthy that I won’t bother listing it here. The images we obtain from each phase are:





To make things more complicated, we can give commands such as “save the current position of the pencil”, “set the angle α to 30o”, “change the segment of line to 5 cm”, “load the last saved position of the pencil”, etc. To make things even more complicated, people, not us, often use recursions, loops and conditional flows. It should be easier to understand when you look at the following example. Pay attention to the different widths, lengths and directions of the branches, and how the program draws the tree: from the root it goes up to draw a big branch, up again to draw smaller branches, then goes back towards the root and draws other big branches from which smaller branches may extend. They represent the change in segment of line, in angle α variables and the above save load functions respectively.


In practice, complex results can be obtained from combining many rules at the same time. These techniques are collected into the Lindenmayer-system, a common method for constructing fractal. Some fine examples generated using the L-system:


  


There are many more systems, but I’m through with this boring theoretical stuff, please google for more information. For a change, we will spend some time on the applications of fractals.

 

Applications – Fractal Arts


Studies on fractals have been conducted over the last century, and their applications are increasing in number. For example, there is now a branch of mathematics called fractal geometry, and seismology, finance, physics, etc. all benefit from it. Thanks to their irregular and complex structure, fractals can be used to represent real world models, which in turn is one of the reasons they are becoming more and more popular in computer science. The most well known applications should be data compression, procedural generation in games and fractal arts. As I don’t feel like talking about difficult stuff, we’ll only concentrate on fractal arts.

 

Fractal art is now a remarkable subject both in arts and computer science. As one of the youngest braches of modern arts, fractal art has an impressive number of practitioners, which is growing exponentially over the last few years. Fractal art in general uses fractal geometry functions to create similar patterns or self replicas. The results are usually rearranged and modified to enhance the final artwork.

 

The artists often make use of softwares to create their pieces, and some even build their own softwares for this purpose. Several of these softwares specialize in generating fractal landscapes, others in drawing artistic pictures. For artists at the level of using tools, they have to decide on their main fractal from the existing software libraries, arrange the fractal patterns and choose the right colors. Practitioners of generating landscapes as well as fractal experts may construct their pieces from scratch, which includes experimenting with their own constructing algorithms and coloring methods - this sometimes earns them the title “algorists”. Let’s look at some illustrations in both cases:


Fractal landscapes

  

Fractal artworks


   

  


I think you now understand why they’re called artists. But that’s not all there is to fractal art. Computer scientists are also studying on fractal music. Though applying algorithms in music is not new, fractal composition is still in its elementary stage. Similar to fractal imaging, composition has softwares level and algorithms level, and the latter can be considered a subdivision of signal processing in computer science. Fractal music usually bears similarities to electronic music due to its nature, which are chaos and unpredictability. However, it can fit in with guitars, pianos and other instruments to create good compositions. Below is a small collection of fractal instrumentals.

   


Fractal as well as fractal art has many opposing points of view. Some think it shallow, some think it deep. Some find it boring, some find it inspiring. Some see a mess, some see the harmony among chaos. What do you think about fractals? How do you feel looking at the images above? Whatever the opinion is, from now on it's a part of your life. Welcome to the world of fractals.


Further information

  • In the computer graphics course in our university, you would have a 2D or 3D assignment as the final project. The 2D project would likely involve generating your own kinds of fractal plants using your own algorithms. In the worst case which never happens before, the 3D project would be about generating landscapes. Nothing is impossible considering the rising popularity of fractals in our fields. Better be prepared :D
  • The most popular softwares for constructing fractals are UltraFractal and Apophyis, in which the latter is freeware. The 3D images in this article are generated from Incendia, an interesting piece of software in development, which is being promoted by the fractal community in Deviantart.
  • Electric Sheep - an online open-source project. The homepage has a large collection of fractal flames created by the free fractal generating screensaver Electric Sheep. When the screensaver is running, it automatically downloads fractals from other sources and, in a way, evolves by adding different algorithms to the animated flames. The community can vote for their favorite animations so that the most beautiful flames stay longer.
  • Media are taken from the following websites:

wikipedia.org

deviantart.com

fractovia.org

fractalmusician.com

donarcher.com

 

And a sample instrumental that comes with the software Fractal Tune Smithy. Some music files are converted from MIDI format.

  • All files are for illustration purpose only. All media are the properties of their respective owners.