Fractal Geometry

IFS with Memory

Graphical Representation

In the standard IFS formalism, all combinations of transformations are allowed. By allowing only certain combinations, we open the possibility of generating many, many more images.

Every combination that is not allowed is forbidden, so it suffices to specify either the allowed or the forbidden combinations.

Note that if T_i₁...T_{i_N} is forbidden, then so is T_{i_a}...T_{i_b} T_i₁...T_{i_N} T_{i_c}...T_{i_d} for all sequences i_a...i_b and i_c...i_d, including empty sequences.

Consequently, the simplest IFS with memory is specified by forbidden pairs, equivalently, by allowed pairs.

For our transformations T₁, T₂, T₃, and T₄, this information can be represented visually by a graph.

The graph has

four vertices, one for each T_i, and

an edge from vertex i to vertex j if T_i can be be followed immediately by T_j.

For example, this graph

specifies that T₁ cannot be followed immediately by T₄, because the graph has no arrow from vertex 1 to vertex 4.

All other combinations are allowed, so all other pairs of edges are connected by arrows.

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