IFS with Memory

T₄ never occurs

If T₄ is never applied, then no points land in the subsquare S₄ with address 4. That is, the upper right subsquare is completely empty.

Because the subsquare S₄ contains no points, the subsquares T₁(S₄) = S₁₄, T₂(S₄) = S₂₄, and T₃(S₄) = S₃₄ contain no points.

Continuing, the subsquares T₁(S₁₄) = S₁₁₄, T₁(S₂₄) = S₁₂₄, T₁(S₃₄) = S₁₃₄, ..., and T₃(S₃₄) = S₃₃₄ contain no points. That is,

if i=4, j=4, or k=4, the subsquare S_ijk contains no points (below left).

Similarly,

the subsquare S_ijkm contains no points if any of i, j, k, or m is 4 (below right).

The result of continuing this process is clear:

if T₄ is never applied, every square whose address contains a 4 is empty.

With this restrction, we see the IFS generates a right isosceles Sierpinski gasket. This is no surprise, because the IFS {T₁, T₂, T₃} generates a right isosceles Sierpinski gasket.

Return to Four Gaskets.