# Mathematical Experiment 14

 Construct these markings on a ruler:
Construct the "snowflake".

These are the basic steps:

 Here is the main part: 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;
 Construct the ``W-curve'': (Reference: J.S. Rohl, Recursion via Pascal, construction with Turbo Basic) 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(0,0,4),color=black,axes=none);