The Thue-Morse binary sequence starts with zero and one successively appends to the existing sequence the boolean compliment of the sequence so far. One interesting feature is that a turtle graphics program can be written to control the turtle by feeding it the successive digits from the sequence so that:
- A zero moves the cursor forward by a step
- A one turns the cursor anti-clockwise by 60 degress (or pi/3 radians)
#include<stdio.h>
#include <math.h>
#define Q (1<<9)
#define M /**/for
#define E(b,a) {\
J q=1^(X+b)[3+Q*\
a];printf("%c%c%\
c",q,q,q);/*//*/}
int /*/*/ typedef
#define W double
#define z (Q<<5)
J;J X[3+(Q *Q)]
,t[z ] ;J main(
){ W o= Q >>1,V=
o/5, a=0; M (1 [X
]=1; X[1] <z;X
[1 ] *=2) for(
X[2] =0;X [2]<
X[1] ;X[2 ]++,
t[X[ 2]+X [1]]
=1^t [X[2 ]]);
for( 0[X] =0;X
[0]< 3;0[ X]++
)for (X[2] =0;X
[2]< z;X[ 2]++
)if( t[X[ 2]])
a-=( M_PI /3.0
);else{o+= cos(a ); V+=
sin(a);X[3+( int)o+Q*(int)V]|=1;}printf(
"P6 %d %d 1 ",Q,Q);M(1[X] =0;X[1]<Q;X[1]
++)M(2[X]=0;X[ 2]<Q;X[2]++)E(2[X],X[1]);
return 0;} /* Colin Ian King 2013 */
gcc koch-binary.c -lm -o koch-binary
./koch-binary | ppmtojpeg > koch-binary.jpg
The original source code can be found here. It contains a 512 x 512 turtle graphics plotter, a Thue-Morse binary sequence generator and a PPM output backend. Anyway, have fun figuring out how it works (it is only mildly obfuscated) and have a great Christmas!
