﻿ The Beauty of Mathematics
21 May, 2013, 15:49:57
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AngelFish
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 « on: 21 July, 2011, 20:11:49 » +4

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 Hidden:

Spoiler for Atrocious Mathematica code to generate the image:
Quote
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]

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:

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 Hidden:

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 Hidden:

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 Hidden:

If anyone has anything else, feel free to link it below.
 « Last Edit: 22 July, 2011, 19:22:48 by Qwerty.55 » Logged

∂²Ψ    -(2m(V(x)-E)Ψ
---  = -------------
∂x²        ℏ²Ψ
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 « Reply #1 on: 21 July, 2011, 20:36:58 » 0

The last pic shows a ship of the "Replicatoren" (german) of Stargate SG1!
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 « Reply #2 on: 21 July, 2011, 21:09:14 » 0

Wow! These are awesome! Thanks, Qwerty!
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 « Reply #3 on: 21 July, 2011, 23:51:57 » 0

The second pic is pretty awesome.
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 « Reply #4 on: 22 July, 2011, 01:15:55 » 0

How do you make pics of the fractals like that--is there a generator somewhere or something?
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 « Reply #5 on: 22 July, 2011, 01:22:07 » 0

I want a 3D Game of Life Generator now!
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Late last night, Quebec was invaded by a group calling themselves, "Omnimaga". Not much is known about these mysterious people except that they all carried calculators of some kind and they all seemed to converge on one house in particular. Experts estimate that the combined power of their fabled calculators is greater than all the worlds super computers put together. The group seems to be holding out in the home of a certain DJ_O, who the Omnimagians claim to be their founder. Such power has put the world at a standstill with everyone waiting to see what the Omnimagians will do...

Wait... This just in, the Omnimagians have sent the UN a list of demands that must be met or else the world will be "submitted to the wrath of Netham45's Lobster Army". Such demands include >9001 crates of peanuts, sacrificial blue lobsters, and a wide assortment of cherry flavored items. With such computing power stored in the hands of such people, we can only hope these demands are met.

In the wake of these events, we can only ask, Why? Why do these people make these demands, what caused them to gather, and what are their future plans...

Spoiler for Misc:

Spoiler for Goosed stuff:
FUSION-TIME GO!
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 « Reply #6 on: 22 July, 2011, 01:23:21 » +1

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:
 1234567891011121314151617181920212223242526272829303132333435363738 FUNCTION decToRom\$ (num AS INTEGER)DIM rom AS STRINGDIM dec AS INTEGERDIM i AS INTEGERDIM j AS INTEGERDIM x AS INTEGERdec = num + 1 - 1 ' Bonus points if you can explain why this is here.IF dec < 1 OR dec > 3999 THEN ERROR 5i = 1DO 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 + 2LOOPdecToRom\$ = romEND FUNCTIONFUNCTION romchar\$ (quark AS INTEGER)romchar\$ = MID\$("IVXLCDM", quark, 1)END FUNCTION
 « Last Edit: 22 July, 2011, 01:28:28 by DrDnar » Logged

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 « Reply #7 on: 22 July, 2011, 01:30:08 » 0

Wow!  Math can be amazing at many, many times.  Those images are truly beautiful!
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 « Reply #8 on: 22 July, 2011, 03:03:13 » 0

@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.
 « Last Edit: 22 July, 2011, 03:04:32 by Qwerty.55 » Logged

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 « Reply #9 on: 22 July, 2011, 04:03:26 » 0

Nope. I put it there because QuickBASIC uses pass-by-reference by default.
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 « Reply #10 on: 22 July, 2011, 06:50:28 » 0

That's (part of) why I love math

And there's another beauty outside the graphs. Aren't equations like eπi+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
 « Last Edit: 22 July, 2011, 06:51:48 by Deep Thought » Logged

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 « Reply #11 on: 22 July, 2011, 19:17:29 » 0

This is a Replikator:

I only know these three types of Replikatoren-ships, but it looks like a ship of them, too!
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MUSIC:
Dragonforce - Fury of the storm
Dragonforce - Through fire and flames
Skillet - Awake and alive
Skillet - Monster
Nelly - Just a dream
Linkin Park - No more Sorrow
Silbermond - An alle Krieger des Lichts
Wise Guys - Das bedeutet Krieg!
Wise Guys - Das war's wert!

GAMES:
Fancy Pants adventures - World 1
- World 2
- Sneak Peak
- World 3 demo
Droid Assault
Age of Empires
Empire Earth

VIDEOS/FILMS:
StarGate SG1
StarTrek
TheAnnoyingOrange
V for Vendetta
War Games (1 + 2)

BOOKS:
Anthony Horowitz - Alex Rider 1: Stormbreaker
- Alex Rider 2: Skelleton Key
- Alex Rider 3: Point Blanc
- Alex Rider 4: Eagle Strike
- Alex Rider 5: Ark Angle
- Alex Rider 6: Scorpia
- Alex Rider 7: Crocodile tears
- Alex Rider 8: Scorpia Rising
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- Still don't have #8
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 « Reply #12 on: 22 July, 2011, 22:23:12 » 0

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:

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It can make you lose
 « Last Edit: 22 July, 2011, 23:08:25 by HOMER-16 » Logged

Spoiler for Things I find interesting:
Spoiler for AI Programming:
Spoiler for Historical warfare/Formations(Warning: Large images):
Phalanx

Testudo

Schiltron

Better version
Terico

Line

Square

Spoiler for OldSig:
Imma Cat! =^_^= (Make this an emoticon!)

Spoiler for Projects!:

Spoiler for IMPORTANT NEWS!:
Late last night, Quebec was invaded by a group calling themselves, "Omnimaga". Not much is known about these mysterious people except that they all carried calculators of some kind and they all seemed to converge on one house in particular. Experts estimate that the combined power of their fabled calculators is greater than all the worlds super computers put together. The group seems to be holding out in the home of a certain DJ_O, who the Omnimagians claim to be their founder. Such power has put the world at a standstill with everyone waiting to see what the Omnimagians will do...

Wait... This just in, the Omnimagians have sent the UN a list of demands that must be met or else the world will be "submitted to the wrath of Netham45's Lobster Army". Such demands include >9001 crates of peanuts, sacrificial blue lobsters, and a wide assortment of cherry flavored items. With such computing power stored in the hands of such people, we can only hope these demands are met.

In the wake of these events, we can only ask, Why? Why do these people make these demands, what caused them to gather, and what are their future plans...

Spoiler for Misc:

Spoiler for Goosed stuff:
FUSION-TIME GO!
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 « Reply #13 on: 22 July, 2011, 23:18:04 » 0

Aw, that's cheating. You actually have to make a pattern that displays that, like the Golly ticker pattern.
 « Last Edit: 22 July, 2011, 23:18:16 by calcdude84se » Logged

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 « Reply #14 on: 22 July, 2011, 23:19:48 » 0

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/
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Spoiler for Things I find interesting:
Spoiler for AI Programming:
Spoiler for Historical warfare/Formations(Warning: Large images):
Phalanx

Testudo

Schiltron

Better version
Terico

Line

Square

Spoiler for OldSig:
Imma Cat! =^_^= (Make this an emoticon!)

Spoiler for Projects!:

Spoiler for IMPORTANT NEWS!:
Late last night, Quebec was invaded by a group calling themselves, "Omnimaga". Not much is known about these mysterious people except that they all carried calculators of some kind and they all seemed to converge on one house in particular. Experts estimate that the combined power of their fabled calculators is greater than all the worlds super computers put together. The group seems to be holding out in the home of a certain DJ_O, who the Omnimagians claim to be their founder. Such power has put the world at a standstill with everyone waiting to see what the Omnimagians will do...

Wait... This just in, the Omnimagians have sent the UN a list of demands that must be met or else the world will be "submitted to the wrath of Netham45's Lobster Army". Such demands include >9001 crates of peanuts, sacrificial blue lobsters, and a wide assortment of cherry flavored items. With such computing power stored in the hands of such people, we can only hope these demands are met.

In the wake of these events, we can only ask, Why? Why do these people make these demands, what caused them to gather, and what are their future plans...

Spoiler for Misc:

Spoiler for Goosed stuff:
FUSION-TIME GO!
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