Pre-previously on Overlord-In-Terms-of-Issues-Around-Engagement-with-the-Über-Feral, I’ve looked at various ways of creating fractals by restricting the moves of a point jumping towards the vertices of a polygon. For example, the point can be banned from jumping towards the same vertex twice in a row. This time, I want to look at fractals created not by restriction, but by compulsion. If the point jumps towards vertex v and then tries to jump towards vertex v again, it will be forced to jump towards vertex v+1 instead, and so on.

You could call v → v+1 a forced increment or finc. So these are finc fractals. In some cases, restriction and compulsion create the same fractals, but I’ve found some new fractals using compulsion. Consider the fractal created by the rule v_[-2]+1, v_[-1] → +0,+1, where the subscripts refer to the history of jumps: v_[-2] is the jump-before-last, v_[-1] is the last jump. If the new vertex, v_[0], chosen is the same as v_[-2]+1 (e.g., v_[0] = 2 = v_[-2]+1 = 1+1), then the forced increment is 0, i.e., the point is allowed to choose that jump. However, if v_[0] = v_[-1], then the forced increment is 1 and the point must jump towards v_[-1]+1.

Here is the fractal in question:

v_[-2]+1, v_[-1] → +0,+1 (black-and-white)

v_[-2]+1, v_[-1] → +0,+1 (colour)

1,0 → +0,+1 (animated)

1,0 → +1,+0 (bw)

1,0 → +1,+0 (col)

1,0 → +1,+0 (anim)

1,0 → +1,+1 (bw)

1,0 → +1,+1 (col)

1,0 → +1,+1 (animated)

0,1 → +2,+1 (anim)

0,1 → +3,+1

1,0 → +0,+1

1,0 → +1,+0

1,1 → +0,+1

1,1 → +1,+2

1,1 → +1,+3

1,1 → +2,+1

1,2 → +0,+3

1,3 → +0,+1

2,2 → +0,+1

But suppose the history of jumps records not actual jumps, but the jumps the point wanted to make instead. In some cases, the jump made will be the same as the jump originally chosen, but in other cases it won’t. Here are some fractals using this method:

0 → +2

0 → +3

2 → +1

2 → +2