Maths is smART

Star Fractals cont.

Tue, 01 Nov 2011 08:11:00 -0400

Here is the last of the star fractals followed by the Matlab code. If you have any questions about the code please ask.

function A = starsfractal(itterations, N, d, x_center, y_center, radius, direction, null_pt)
hold on
start = (N-2)*pi/(2*N) + mod(N,2)*direction*pi/N;
t = linspace(start,start+2*pi,N+1);
t(N+1)=[];
x = radius*cos(t);
y = radius*sin(t);
for i=1:N
    X = [];
    Y = [];
    for j=1:N+1
        X = [X x(mod(i+j*d,N)+1)];
        Y = [Y y(mod(i+j*d,N)+1)];
    end
    set(gca, ‘ColorOrder’, [0.2 0.4 0.3]);
    plot(X + x_center,Y + y_center)
end

if itterations ~= 1
    star_pos = 1:N;
    if null_pt ~= 0
        star_pos(null_pt) = [];
    end
    for i=star_pos
        j = mod(i+floor(N/2)+direction*mod(N,2),N);
        if j == 0
            j = j+N;
        end
        starsfractal(itterations - 1, N,d,4*x(i)/3 + x_center, 4*y(i)/3 + y_center, radius/3, mod(direction + 1,2),j);
    end
end

There are so many inputs because of the recursive construction of the fractals. The only things you need to worry about is itterations, N and d. You should set the other values as default:

x_center = 0
y_center = 0
radius = 1
direction = 1
null_pt = 0

How cool is this star fractal :)

Thu, 27 Oct 2011 16:30:53 -0400

How cool is this star fractal :)

Star Fractal

Tue, 25 Oct 2011 18:25:00 -0400

It is quite hard to define what a fractal is… Here is what wikipedia says:

A fractal has been defined as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,”a property called self-similarity.

As well as fractals appearing in nature (which I think is quite amazing), there are many famous “man-made” fractals which are created by some given algorithm. One example of these is the star fractal. Basically you draw some star polygon and at each point draw the same star polygon ‘the other way up’ of some smaller propotion…. continue until you can’t see your stars turn to dots :).

I didn’t want to make the picture myself so I decided to get matlab to do it for me, I have used the simple pentagram as my star-polygon but I have coded the program so I can easily use any star-polygon:

I will post some more star-fractals tomorrow as well as the code if you would like it. Please enjoy :)

Ordered Squiggliness

Mon, 24 Oct 2011 12:42:49 -0400

I hope you enjoy the following pictures created in matlab. Although it seems these pictures are disordered they were created with only the functions sin,cos,exp and sqrt. No randomness was used!

A ball of string

Rhubarb Crumble

Pain Wheel

Name these shapes

Wed, 19 Oct 2011 16:56:00 -0400

I like creating new shapes using some algorithm. I don’t think the following shapes have a name so please help me come up with one, alternatively if they already have a name let know.

I have created the following shapes using this algorithm, given an integer N:

Place N points uniformly around a circle (starting from the top of the circle).
Connect each point with a line to the top and bottom of the circle.

Examples:

N = 25

N = 50

N = 100

N = 1000

Question: What are the angles created by two lines meeting at an end point?

Everyone was too smart for me and guessed the right family of...

Tue, 18 Oct 2011 15:31:00 -0400

Everyone was too smart for me and guessed the right family of distributions, that is the beta distribution. How about these? If you don’t know just enjoy the picture :)

These plots came from the same family of probability...

Mon, 17 Oct 2011 16:26:50 -0400

These plots came from the same family of probability distributions. Can you tell me which one it is?

A pretty picture!

Wed, 31 Aug 2011 03:07:00 -0400

I hope you like this picture, this picture was wholey created using matlab! Maybe not the most productive thing to use matlab for but it was fun and I like the end result:

If you would like to see more pictures like this just give me a scene or scenario and I will have a go. Thanks! If any of you wanted the code here it is:

%sky
clf
x = [-3:0.1:13];
hold on
for i=1:0.2:13
    set(gca, ‘ColorOrder’, [rand*50 rand*55+200 rand*100]./255);
    set(gca, ‘ColorOrder’, [rand*50 rand*100 rand*55+200]./255);
    plot(x,i);
end

%sun
nopoints = 20;
x = linspace(-1.88,1.88,nopoints);
dt = 0.05;
n = 1/dt;
theta = 2*pi*dt;
A = [cos(theta) -sin(theta); sin(theta) cos(theta)];
y = cos(x);
for i=1:n
    for j=1:nopoints
        v = [x(j);y(j)];
        vnew = A*v;
        x(j) = vnew(1);
        y(j) = vnew(2);
    end
    set(gca, ‘ColorOrder’, [rand*50+205 rand*50+205 rand*50]./255);
    plot(x+10,y+10)
end

%tree trunk
x1 = [-1:0.01:-0.2];
x2 = [0.2:0.01:1];
set(gca, ‘ColorOrder’, [139 69 19]./255);
for i=0:0.1:0.4
    plot(x1+i,0.2./(x1.^2))
end
for i=0:0.1:0.4
    plot(x2-i,0.2./(x2.^2))
end

%tree top
t = linspace(0,16*pi,1000);
for i = 0.5:0.1:2
    x = (2*cos(t)+i*cos(8*t))./2;
    y = (sin(t)+i*sin(8*t))./2+5;
    set(gca, ‘ColorOrder’, [rand*100 rand*50+205 rand*50]./255);
    plot(x,y)
end

%grass
x = [-3:0.1:13];
for i=-4:0.05:1
    set(gca, ‘ColorOrder’, [rand*50 rand*55+200 rand*100]./255);
    plot(x,i);
end

%rainbow
n = 140;
ColourMatrix = zeros(n+2,3);
for i=1:n/7
    ColourMatrix(i,:) = [255 153*7*i/n 0]./255;
    ColourMatrix(i+n/7,:) = [255 (255-153)*7*i/n+153 0]./255;
    ColourMatrix(i+2*n/7,:) = [-255*7*i/n+255 255 0]./255;
    ColourMatrix(i+3*n/7,:) = [-255*7*i/n+255 -255*7*i/n+255 255*7*i/n]./255;
    ColourMatrix(i+4*n/7,:) = [0 65*7*i/n (106-255)*7*i/n+255]./255;
    ColourMatrix(i+5*n/7,:) = [143*7*i/n -65*7*i/n+65 (255-106)*7*i/n+106]./255;
    ColourMatrix(i+6*n/7,:) = [(255-143)*7*i/n+143 255*i/n 255]./255;
end
t = linspace(0,pi,n);
for r=linspace(5,6,n)
    x = (11-r)/2*cos(t)+9;
    y = (11-r)/2*sin(t)+0.8;
    set(gca, ‘ColorOrder’,ColourMatrix(floor((r-5)*n+1),:));
    plot(x,y)
end
hold off

Magic Eye I got alot of good response from the picture I created...

Tue, 30 Aug 2011 07:52:00 -0400

Magic Eye

I got alot of good response from the picture I created on Gnuplot I decided to try and make another one. Again this was a simple code produced with C and copied into Gnuplot terminal, it kind of looks like one of those magic eye pictures don’t you think? Perhaps if you stare at it long enough it will spell something out to you… If so let me know :)

Gnuplot

Sun, 28 Aug 2011 14:52:00 -0400

I have only recently started using Gnuplot (literally only for a few weeks) but combined with C it is a powerful way of creating nice images. I used this simple code in C:

#include <stdio.h>

int main()
{
    int i;
    printf(“plot [0:10][1:9] \”);
    for(i = 0; i < 200 ; i++)
        printf(“sin(x)+(%f) with filledcurve x2, \\n”, (double) i/20);

    printf(“sin(x)+10 with filledcurve x2”);
    return(0);
}

Then I copied the output into Gnuplot and got the following result:

Obviously very simple but I will try to master Gnuplot and post some really cool pictures!

Double Rainbow Double rainbow all the way across the sky!

Tue, 23 Aug 2011 15:05:00 -0400

Double Rainbow

Double rainbow all the way across the sky!

RainbowThis was actually quite a challenge to do, I had to...

Tue, 23 Aug 2011 02:12:00 -0400

Rainbow
This was actually quite a challenge to do, I had to create a colour matrix that created a gradual change to each colour in the rainbow. Here is the code:

clf
hold on
n = 140;
ColourMatrix = zeros(n+2,3);
for i=1:n/7
    ColourMatrix(i,:) = [255 153*7*i/n 0]./255;
    ColourMatrix(i+n/7,:) = [255 (255-153)*7*i/n+153 0]./255;
    ColourMatrix(i+2*n/7,:) = [-255*7*i/n+255 255 0]./255;
    ColourMatrix(i+3*n/7,:) = [-255*7*i/n+255 -255*7*i/n+255 255*7*i/n]./255;
    ColourMatrix(i+4*n/7,:) = [0 65*7*i/n (106-255)*7*i/n+255]./255;
    ColourMatrix(i+5*n/7,:) = [143*7*i/n -65*7*i/n+65 (255-106)*7*i/n+106]./255;
    ColourMatrix(i+6*n/7,:) = [(255-143)*7*i/n+143 255*i/n 255]./255;
end
t = linspace(0,pi,n);
for r=linspace(5,6,n)
    x = (11-r)*cos(t);
    y = (11-r)*sin(t);
    set(gca, ‘ColorOrder’, ColourMatrix(floor((r-5)*n+1),:));
    plot(x,y)
end
hold off

I hope I explain well enough, if you ever want to ask me something you can email me on mathsissmart@gmail.com

More Brownian Motion

Sat, 20 Aug 2011 18:57:00 -0400

I generated Brownian motion in one dimension and changed the colour matrix to obtain different textures. Here are some of the results:

Coolmathguy

Tue, 16 Aug 2011 16:06:00 -0400

Hey guys! I would like to thank you again for all the support you have showed me. I know this is still a very small blog but I am proud of the interest that I have gotten so far. I have just been posted on the website http://coolmathguy.com/ which is a big deal for me as I have been visiting this website regularly for a while now. You can find the post at the following link: http://coolmathguy.com/little-math-art-tumblr

Now I just wanted to show you one more picture I created using the code from “Around and around” which I posted yesterday. Am sorry for the it being too repeatitive but I promise something more interesting coming soon! Thanks again :)

Around and around! cont.

Mon, 15 Aug 2011 18:33:00 -0400

Some more pictures using the code that I posted yesterday.

exp(x)

cos^2(x)

And the following one which is my favourite:

cos(x)

Thank you all for the support and encouragement :)

Around and around!

Sun, 14 Aug 2011 15:35:00 -0400

About a week ago I blogged about randomness and one of the pictures I posted looked like this:

This was created using the following code:

nopoints = 20;
x = linspace(-1,1,nopoints);
dt = 0.005;
n = 1/dt;
theta = 2*pi*dt;
A = [cos(theta) -sin(theta); sin(theta) cos(theta)];
y = x;
clf
hold on
for i=1:n
    for j=1:nopoints
        v = [x(j);y(j)];
        vnew = A*v;
        x(j) = vnew(1);
        y(j) = vnew(2);
    end
    set(gca, ‘ColorOrder’, rand(1,3));
    plot(x+10,y+10)
end
hold off

Granted this is a bit complicated for the effected created but I wanted to change the 7th line to another function and just by that simple change create more interesting pictures. Well that is what I have done, and here are a few of the consequences:

y = x^3

y = tan(x)

I will post more tomorrow, including my favourite! For now, what function do you think I used for this picture:

Diamonds

Sun, 14 Aug 2011 09:03:00 -0400

I am going to be really busy for the next 2 weeks so won’t be able to post anything too amazing but I have some really exciting projects that I have had to put on hold. Anyways, here are a few cool pictures:

points=linspace(0,2*pi,200);clfhold onfor i = points ...

Thu, 11 Aug 2011 21:16:00 -0400

points=linspace(0,2*pi,200);
clf
hold on
for i = points
    t=linspace(0,2*pi,100);
    x = cos(i)+sin(t);
    y = sin(i)+cos(t);
    set(gca, ‘ColorOrder’, (rand(1,3)+1)./2);
    plot(x,y)
end
    x = 2*sin(t);
    y = 2*cos(t);
    set(gca, ‘ColorOrder’, [0 0 0]);
    p = plot(x,y)
    set(p,’Color’,’black’,’LineWidth’,2)
hold off

A simulation made in Matlab of 10 single realiations of a...

Thu, 11 Aug 2011 13:17:02 -0400

A simulation made in Matlab of 10 single realiations of a two-dimensional Wiener process.

Brownian motion/Wiener Process

Wed, 10 Aug 2011 19:41:59 -0400

I don’t know much about the history of this subject. All I know (or what I think I know) is that Robert Brown was a botanist who observed the random movement of pollen grains in water. The continuous and jerky movement of such a particle is what we describe as Brownian motion and the mathematical representation of this is called the Wiener process in honour of Norbert Wiener.

The cool thing about the Wiener process is that it is everywhere continuous but nowhere differentiable. Although a computer could never totally simulate brownian motion I like to create pictures using the idea of Brownian motion.

The trajectories of 10 particles in two dimensions starting from the origin:

clf
t = 2.0;
dt = 0.005;
n = floor(t/dt);
ntraj = 20;
T = linspace(0,2*pi,100);
circle_x = 10*sin(T)
circle_y =
x = zeros(ntraj,n+1);
y = zeros(ntraj,n+1);
tspan = [0:dt:t];
for i = 1:n
    x(:,i+1) = x(:,i) + sqrt(dt)*randn(ntraj,1);
    y(:,i+1) = y(:,i) + sqrt(dt)*randn(ntraj,1);
end
hold on
for i=1:ntraj
    set(gca, ‘ColorOrder’, rand(1,3));
    plot(x(i,:),y(i,:))
end
hold off

I will be posting an animation of 10 particles of a two dimension Wiener process.