The Beauty of Mathematics

[AngelFish]

As most of us are know, mathematics can be very brutal and ugly, especially when dealing with algebraic equations. Here are a few of its prettier sides.

Warning: the images in spoilers are large.

This image was generated during a search for a more efficient way to compute A Mod B. It's a plot of the error between the true answer and my approximation over all possible 16 bit operands. I'm not sure why the resulting graph is so complex given the simplicity of the underlying function, but the remarkable self-similarities and other fractal qualities make it interesting.

Spoiler For Spoiler:

Spoiler For Atrocious Mathematica code to generate the image:

t=2^8;
TestMod[k_,b_,y_,x_]:=Mod[k,b]-Mod[k-y,b-x];
Array1=Array[0&,t^2];
Array2=Array[0&,t^2];
Array3=Array[0&,t^2];
Array4=Array[0&,t^2];
j=0;
For[b = 1, b < t, b++,

For[k = 1, k < t, k++,
y=k;
x=b;
x=BitOr[x,BitShiftRight[x,1]];
x=BitOr[x,BitShiftRight[x,2]];
x=BitOr[x,BitShiftRight[x,4]];
x=BitOr[x,BitShiftRight[x,8]];
x=BitOr[x,BitShiftRight[x,16]];
x=x-BitShiftRight[x,1];

y=BitOr[y,BitShiftRight[y,1]];
y=BitOr[y,BitShiftRight[y,2]];
y=BitOr[y,BitShiftRight[y,4]];
y=BitOr[y,BitShiftRight[y,8]];
y=BitOr[y,BitShiftRight[y,16]];
y=y-BitShiftRight[y,1];

Array1[[t*(b-1)+k]]=Mod[k,b];
Array2[[t*(b-1)+k]]=Mod[k-y,b-x];
Array3[[t*(b-1)+k]]=TestMod[k,b,y,x];
If[TestMod[k,b,y,x]==0,j=j+1]
]
]

ListPlot[Array3]

External link: http://img.removedfromgame.com/imgs/graph.png


The next picture is a strange attractor that belongs to a family of functions known as Lorenz equations. This particular family is characterized by their extreme sensitivity to initial conditions. A change of one in a hundred parts can result in a completely different system.

Spoiler For Lorenz attractor:

External link: http://img.removedfromgame.com/imgs/Lorenz_fractal.bmp


Here is an example of such a change. A single constant in the equations was changed to produce this graph from the previous one.

Spoiler For Spoiler:

External link: http://img.removedfromgame.com/imgs/Lorenz_lines.bmp


Another group of functions with interesting graphs are the one dimensional elementary cellular automata. In addition to their interesting computational properties (several of them are Turing-equivalent), some of them are chaotic to the point that they can safely be used for cryptographic grade random number generators. One particularly well known automaton is Rule 110 (following Wolfram's notation). The picture, which started from random initial conditions, clearly shows the propagating structures that form the basis of this rule's universal computation.

Spoiler For Spoiler:

External link: http://img.removedfromgame.com/imgs/Rule_110.png

Cellular automata can occupy more than one dimension, though. Here's a model of a two dimensional cellular automaton known as Conway's life rendered over time.

Spoiler For Spoiler:

External link: http://www.frank-buss.de/automaton/symmetric_rabbits.gif

If anyone has anything else, feel free to link it below.

[p2]

The last pic shows a ship of the "Replicatoren" (german) of Stargate SG1!

[ZippyDee]

Wow! These are awesome! Thanks, Qwerty!

[Anima]

The second pic is pretty awesome.

[Darl181]

How do you make pics of the fractals like that--is there a generator somewhere or something?

[Scipi]

I want a 3D Game of Life Generator now!

[DrDnar]

Make a plot of the length of all Roman numerals less than some value and you get a bell curve.

EDIT: Code for the lazy:

Code: [Select]

FUNCTION decToRom$ (num AS INTEGER)
DIM rom AS STRING
DIM dec AS INTEGER
DIM i AS INTEGER
DIM j AS INTEGER
DIM x AS INTEGER

dec = num + 1 - 1 ' Bonus points if you can explain why this is here.


IF dec < 1 OR dec > 3999 THEN ERROR 5

i = 1
DO WHILE dec / 10 > 0
    x = dec - INT(dec / 10) * 10 ' Mod is for losers.
    dec = INT(dec / 10)
    IF x >= 1 AND x <= 3 THEN
        FOR j = 1 TO x
            rom = romchar(i) + rom
        NEXT j
    END IF
    IF x = 4 THEN rom = romchar(i) + romchar(i + 1) + rom
    IF x = 5 THEN rom = romchar(i + 1) + rom
    IF x >= 6 AND x <= 8 THEN
        FOR j = 1 TO x - 5
            rom = romchar(i) + rom
        NEXT j
        rom = romchar(i + 1) + rom
    END IF
    IF x = 9 THEN rom = romchar(i) + romchar(i + 2) + rom
    i = i + 2
LOOP
decToRom$ = rom
END FUNCTION

FUNCTION romchar$ (quark AS INTEGER)
romchar$ = MID$("IVXLCDM", quark, 1)
END FUNCTION

[apcalc]

Wow!  Math can be amazing at many, many times.  Those images are truly beautiful!

[AngelFish]

@Darl: I used Chaosope to generate the second and third pictures. There are a lot of other attractor/fractal generating packages, but I find that one produces nice images without too much fuss. The first and fourth pictures were generated using Mathematica, which I don't recommend purchasing unless you have some means of obtaining it for free/small cost (generally schools or piracy).

I want a 3D Game of Life Generator now!

Here you go. Requires Golly, Blender, and Python.

@DrDnar: I presume that the Dec+1-1 somehow detects/fixes integer overflow? Anyway, while the mathematics behind Gaussians are very elegant and pretty, I made a conscious choice not to include them because they don't generally look terribly impressive.

[DrDnar]

Nope. I put it there because QuickBASIC uses pass-by-reference by default.

[Deep Toaster]

That's (part of) why I love math

And there's another beauty outside the graphs. Aren't equations like e^πi+1=0 just amazing to think about? Even with all those proofs, many of which aren't even that advanced, I still can't wrap my head around the idea.

The last pic shows a ship of the "Replicatoren" (german) of Stargate SG1!

lol

I want a 3D Game of Life Generator now!

Plot three dimensions against time

[p2]

This is a Replikator:

















I only know these three types of Replikatoren-ships, but it looks like a ship of them, too!

[Scipi]

Replicators were the awesomest enemies in the series. Even the Ori would have trouble with them.

Here's something else the Game of Life can do:

Spoiler For Spoiler:

It can make you lose

[calcdude84se]

Aw, that's cheating. You actually have to make a pattern that displays that, like the Golly ticker pattern.

[Scipi]

I don't have to, I have a program that does it for me

http://tlrobinson.net/blog/2009/02/07/game-of-life-generator/