Mathematical Experiment 11

The "W" curve is constructed recursively as thus:   stage 0:  stage 1:  stage 2:  stage 3:  stage 4:

> restart;

> Z:=proc(n) 2^(n+1) end;

> U:=proc(x,y,n) if n=0 then [[x,y+1]] else [op(U(x,y,n-1)),op(R(x,y+Z(n)-3,0)),op(U(x+1,y+Z(n)-3,0)),op(R(x+1,y+Z(n)-2,n-1)),op(U(x+Z(n)-2,y+Z(n)-2,0)), op(L(x+Z(n)-2,y+Z(n)-1,n-1)),op(U(x+1,y+Z(n)-1,0)),op(L(x+1,y+Z(n),0)),op(U(x,y+Z(n),n-1))]; fi; end;

> Dn:=proc(x,y,n) if n=0 then [[x,y-1]] else [op(Dn(x,y,n-1)),op(L(x,y-Z(n)+3,0)),op(Dn(x-1,y-Z(n)+3,0))
,op(L(x-1,y-Z(n)+2,n-1)),op(Dn(x-Z(n)+2,y-Z(n)+2,0)),op(R(x-Z(n)+2,y-Z(n)+1,n-1)),op(Dn(x-1,y-Z(n)+1,0)), op(R(x-1,y-Z(n),0)),op(Dn(x,y-Z(n),n-1))]; fi; end;

> R:=proc(x,y,n) if n=0 then [[x+1,y]] else [op(R(x,y,n-1)),op(Dn(x+Z(n)-3,y,0)),op(R(x+Z(n)-3,y-1,0)),op(Dn(x+Z(n)-2,y-1,n-1)), op(R(x+Z(n)-2,y-Z(n)+2,0)),op(U(x+Z(n)-1,y-Z(n)+2,n-1)),op(R(x+Z(n)-1,y-1,0)), op(U(x+Z(n),y-1,0)),op(R(x+Z(n),y,n-1))];fi;end;

> L:=proc(x,y,n) if n=0 then [[x-1,y]] else [op(L(x,y,n-1)),op(U(x-Z(n)+3,y,0)),op(L(x-Z(n)+3,y+1,0))

,op(U(x-Z(n)+2,y+1,n-1)),op(L(x-Z(n)+2,y+Z(n)-2,0)),op(Dn(x-Z(n)+1,y+Z(n)-2,n-1)),op(L(x-Z(n)+1,y+1,0)), op(Dn(x-Z(n),y+1,0)),op(L(x-Z(n),y,n-1))];fi;end;

> P:=proc(n) 2^(n+2) end;

> W:=proc(x,y,n) [[x,y],op(U(x,y,n)),op(R(x,y+P(n)-3,0)),op(U(x+1,y+P(n)-3,0)),op(R(x+1,y+P(n)-2,n)), op(Dn(x+P(n)-2,y+P(n)-2,0)),op(R(x+P(n)-2,y+P(n)-3,0)),op(Dn(x+P(n)-1,y+P(n)-3,n)),op(L(x+P(n)-1,y,0)), op(Dn(x+P(n)-2,y,0)),op(L(x+P(n)-2,y-1,n)),op(U(x+1,y-1,0)),op(L(x+1,y,0))] end;

> plot(W(0,0,3),color=black,axes=none,scaling=constrained);

The snowflake curve is constructed recursively as thus:   stage 0: stage 1: stage 2: stage 3: stage 4:

> restart;

> snow:=proc(n,lx,ly,rx,ry) local mlx, mly, mrx, mry, kx, ky; mlx:=(2*lx+rx)/3;mly:=(2*ly+ry)/3;mrx:=(lx+2*rx)/3;mry:=(ly+2*ry)/3;kx:=(mlx+mrx)/2+3^(1/2)*(mly-mry)/2;ky:=(mly+mry)/2+3^(1/2)*(mrx-mlx)/2;if n=0 then [[lx,ly],[rx,ry]] else [op(snow(n-1,lx,ly,mlx,mly)),op(snow(n-1,mlx,mly,kx,ky)),op(snow(n-1,kx,ky,mrx,mry)),op(snow(n-1,mrx,mry,rx,ry))] fi end;

> j:=4;

> a:=snow(j,cos(2*Pi/3),sin(2*Pi/3),1,0):

> b:=snow(j,1,0,cos(2*Pi/3),-sin(2*Pi/3)):

> c:=snow(j,cos(2*Pi/3),-sin(2*Pi/3),cos(2*Pi/3),sin(2*Pi/3)):

> plot([op(a),op(b),op(c)],color=black,axes=none,scaling=constrained);

The Sierpinski Triangle is constructed recursively as thus: satge 1:  stage 2: stage 3:  stage 4: stage 5:

> restart;

> s:=proc(n,a,b,c) local k,mab,mbc,mca; global t; k:=n; while k>0 do k:=k-1;mab:=(a+b)/2; mbc:=(b+c)/2;mca:=(c+a)/2; s(k,a,mab,mca); s(k,b,mbc,mab); s(k,c,mbc,mca);t:=[op(t),[mab,mbc,mca,mab]]; od;end;

> t:=[];

> s(4,[0,1],[1,0],[0,0]):

> plot(t,color=black,axes=none,scaling=constrained);