IFS with Memory

Slanted parallel lines

Taking the fixed points at the ends of an edge of the square gives the horizontal or vertical lines already studied.
The only other option is to take the fixed points at the ends of a diagonal of the square. For example, suppose we take 3 and 2.
To obtain parallel lines we have two choices for the images of these fixed points:
1(3) and 4(2)
As before, we look at the length 2 address squares through which the lines must pass.
and obtain
3 → 3, 4 → 3, 1 → 4, 2 → 4, 3 → 1, 4 → 1, 1 → 2, and 2 → 2
Here are the table, graph, and plot. Indeed, we obtain two lines.
   
Exercise 1 How many pairs of slanted lines can be generated in this way? Try it first before consulting the answers.
Exercise 2 Can either of these lines be generated independently of the other? Try it first before consulting the answer.

Return to Line pairs from a fixed point and its image.