Notes

Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints


Non-periodic pattern [forced by 2D constraint]

The color at position x,y in the pattern is given by

a[x_,y_]:=Mod[y+1, 2]/;x+y>0 a[x_,y_]:=0/;Mod[x+y,2]==1 a[x_, y_] :=Mod[Floor[(x-y) 2^((x + y - 6)/4)], 2] /; Mod[x + y, 4] == 2 a[x_,y_]:=1-Sign[Mod[x-y+2,2^((-x-y+8)/4)]]

The origin of the x,y coordinates is the only freedom in this pattern. The nested structure is like the progression of base 2 digit sequences shown on page 117. Negative numbers are effectively represented by complements of digit sequences, much as in typical practical computers. With the procedure described above for finding patterns that satisfy a constraint, generating the pattern shown here is straightforward once the appropriate constraint is identified.

From Stephen Wolfram: A New Kind of Science [citation]