Tweetable Mathematical Art: Generating Images with Compact C++ Functions

Hi everyone. In computer graphics a color is usually described by three integers R, G, B in the range 0‑255. Every pixel on the screen can therefore be represented by an RGB triple.

The question posed by the "Tweetable Mathematical Art" competition is: if a program automatically fills each pixel, what picture will it produce?

The competition, hosted on StackExchange, asks participants to write three C++ functions RD , GR and BL (for red, green and blue). Each function must be no longer than 140 characters, accept two integer parameters i and j (0 ≤ i, j ≤ 1023), and return an integer 0‑255 representing the colour component at pixel (i, j). The functions are then plugged into a driver program that generates a 1024×1024 PPM image.

Driver program (unchanged apart from minor tweaks):

#include <iostream>
#include <cmath>
#include <cstdlib>
#define DIM 1024
#define DM1 (DIM-1)
#define _sq(x) ((x)*(x))
#define _cb(x) abs((x)*(x)*(x))
#define _cr(x) (unsigned char)(pow((x),1.0/3.0))

unsigned char GR(int,int);
unsigned char BL(int,int);
unsigned char RD(int i,int j){ /* YOUR CODE HERE */ }
unsigned char GR(int i,int j){ /* YOUR CODE HERE */ }
unsigned char BL(int i,int j){ /* YOUR CODE HERE */ }
void pixel_write(int,int);
FILE *fp;
int main(){
    fp = fopen("MathPic.ppm","wb");
    fprintf(fp, "P6\n%d %d\n255\n", DIM, DIM);
    for(int j=0;j
Below are several striking artworks created for the contest, together with the source code that produced them.
Martin Büttner’s first piece:
unsigned char RD(int i,int j){
    return (char)(_sq(cos(atan2(j-512,i-512)/2))*255);
}
unsigned char GR(int i,int j){
    return (char)(_sq(cos(atan2(j-512,i-512)/2-2*acos(-1)/3))*255);
}
unsigned char BL(int i,int j){
    return (char)(_sq(cos(atan2(j-512,i-512)/2+2*acos(-1)/3))*255);
}
A second, higher‑ranking work by the same author uses recursion and static storage:
unsigned char RD(int i,int j){
    #define r(n)(rand()%n)
    static char c[1024][1024];
    return !c[i][j]?c[i][j]=!r(999)?r(256):RD((i+r(2))%1024,(j+r(2))%1024):c[i][j];
}
unsigned char GR(int i,int j){
    static char c[1024][1024];
    return !c[i][j]?c[i][j]=!r(999)?r(256):GR((i+r(2))%1024,(j+r(2))%1024):c[i][j];
}
unsigned char BL(int i,int j){
    static char c[1024][1024];
    return !c[i][j]?c[i][j]=!r(999)?r(256):BL((i+r(2))%1024,(j+r(2))%1024):c[i][j];
}
Manuel Kasten contributed a Mandelbrot‑set zoom, generated with the following compact functions:
unsigned char RD(int i,int j){
    double a=0,b=0,c,d,n=0;
    while((c=a*a)+(d=b*b)<4 && n++<880){
        b=2*a*b+j*8e-9-.645411; a=c-d+i*8e-9+.356888;
    }
    return 255*pow((n-80)/800,3.);
}
unsigned char GR(int i,int j){
    double a=0,b=0,c,d,n=0;
    while((c=a*a)+(d=b*b)<4 && n++<880){
        b=2*a*b+j*8e-9-.645411; a=c-d+i*8e-9+.356888;
    }
    return 255*pow((n-80)/800,.7);
}
unsigned char BL(int i,int j){
    double a=0,b=0,c,d,n=0;
    while((c=a*a)+(d=b*b)<4 && n++<880){
        b=2*a*b+j*8e-9-.645411; a=c-d+i*8e-9+.356888;
    }
    return 255*pow((n-80)/800,.5);
}
Another style relies on static variables to drive the colour evolution without using the coordinates at all:
unsigned char RD(int i,int j){
    static double k; k+=rand()/1./RAND_MAX; int l=k; l%=512; return l>255?511-l:l;
}
unsigned char GR(int i,int j){
    static double k; k+=rand()/1./RAND_MAX; int l=k; l%=512; return l>255?511-l:l;
}
unsigned char BL(int i,int j){
    static double k; k+=rand()/1./RAND_MAX; int l=k; l%=512; return l>255?511-l:l;
}
githubphagocyte posted a diffusion‑limited‑aggregation based generator that squeezes three functions into the 140‑character limit by sharing macros:
unsigned char RD(int i,int j){
    #define D DIM
    #define M m[(x+D+(d==0)-(d==2))%D][(y+D+(d==1)-(d==3))%D]
    #define R rand()%D
    #define B m[x][y]
    return (i+j)?256-(BL(i,j))/2:0;
}
unsigned char GR(int i,int j){
    #define A static int m[D][D],e,x,y,d,c[4],f,n;
    if(i+j<1){for(d=D*D;d;d--){m[d%D][d/D]=d%6?0:rand()%2000?1:255;}
    for(n=1
    return RD(i,j);
}
unsigned char BL(int i,int j){
    A;n;n++){x=R;y=R; if(B==1){f=1;for(d=0;d<4;d++){c[d]=M;f=f
2){B=f-1;}else{++e%=4;d=e; if(!c[e]){B=0;M=1;}}}}}
    return m[i][j];
}
Eric Tressler demonstrated a logistic‑map based Feigenbaum bifurcation picture, again using macro tricks to stay within the size limit:
unsigned char RD(int i,int j){
    #define A float a=0,b,k,r,x
    #define B int e,o
    #define C(x) x>255?255:x
    #define R return
    #define D DIM
    R BL(i,j)*(D-i)/D;
}
unsigned char GR(int i,int j){
    #define E DM1
    #define F static float
    #define G for(
    #define H r=a*1.6/D+2.4;x=1.0001*b/D
    R BL(i,j)*(D-j/2)/D;
}
unsigned char BL(int i,int j){
    F c[D][D]; if(i+j<1){A;B;G;a
D/2){e=a;o=(E*x);c[e][o]+=0.01;}}}}}
    R C(c[j][i])*i/D;
}
These examples show how a few dozen characters of C++ can produce surprisingly rich visual patterns, ranging from trigonometric colour maps to classic fractals. Feel free to copy the snippets and experiment with your own ideas.