Talk:Complexity
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Complexity as it relates to cryptography is actually really interesting, although mathematically mind-boggling. Fleshing out that side-topic would be particularly cool. -- rack
Aren't fractals complex math? -- Amw
- I have no idea how cryptography works mathematically. I know there's some serious calculus involved. I also don't know if fractals are all that complex. I think they're a relatively simple algorithm found in nature. Webfork
Yes, fractals are coming from complex math. They are visualisations of equations with complex numbers. Please mind that 'complex' is not an expression for a romantic view of 'something terribly complicated', but a very well defined mathematical concept. In hackerspeak: complex numbers are an enhancement to floats in a similar way as floats are an enhancement to integers.
With integers you can not express the term 'squareroot of -1'. There is no integer that will render a negative number if squared. Complex numbers do exactly that. sqrt(-1) = i. Each complex number has a real part (a float) an an imaginative part (another float). The notation is like this (5.38 + 7.7i).
With these two floats in one complex number it is possible to make a point on an area/canvas. So integers are regular dots on a line, floats can be anywhere on that line, but complex numbers have two dimensions. Just as floats completely include the integers, complex numbers completely include floats. The line of floats is completely IN the area of complex numbers. It is the horizontal null line of a cartesian graph system.
Fractals are created like this:
- Take a formula which uses complex numbers.
- Use one point from the area (= one complex number) and put it into the formula.
- Then take the result and put it into the formula again.
- Repeat a couple of times (like 50 for example).
- Take the result and map it to a color code (actually you would take the abs of the complex number which is the equivalent of the length of a dash from (0 + 0i) to the point where the result is).
- Print this color at the place where the complex number was (the one which you put into the equation in step 2.).
- Do this for all the points on the screen/area and enjoy the patterns that emerge.
There is a class of fractals which are created different. These are called snowfake curves or hilbert curves. They are constructed from a single line by applying a modification rule to parts of the line over and over again. Example: Draw a line and divide it into three parts.
<pre>
---- ----
</pre>
Now take the middle part and make a square out of it:
<pre>
|
|
|
|
</pre> Repeat this with all the new part lines you have.
The interresting thing about these critters is that they are not lines any more. This is meant in a mathematical sense. They do not have the one dimension of a line, because they are partly outside of the line. They 'live' in the area (dimension 2), but yet they are not an area. Therefore their dimension must be a broken number, like 1.735 or such. This number is a broken number showing that this thingie is a fractal (sic!). The value of this number is a measurement of the complexity of the construct.
Since infinity is not a number (Please see http://www.kuro5hin.org/story/2003/6/3/95744/71866 for a good article about this issue), something like infinite complexity is bullshit in a mathematical sense.
On a personal sidenote I have to admit that I hate it when people, who have no or little idea what they are talking about, use well defined terms to describe the funny tickling in the back of their brain, which they felt when a some point they actually thought about something they could not comprehend or express with other words... Sorry.
References and further reading here: http://www.brint.com/Systems.htm
Cryptography by the way has little to do with fractals and complex numbers, even though it uses complex (in terms of complicated) calculus. Cryptography uses calculations on your information which are reversible, but where the reverse process is very difficult to achieve unless you have the key. Public key encryption is built on the experience/assumption that it is very difficult to divide large numbers into prime factors when this large number is a product of two large prime numbers. Multipyling two numbers is very very easy, but guessing which numbers you have multiplied only from the result is very difficult unless you already have one of the two primes. Think of the one of those two prime numbers as your private key and the result of the multiplication as your public key (very simplified).
-jsilence
To discuss - atrophy, entropy, syntropy, apostophy. - ABliss
