[tomadlfa] period-doubling binary tree fractal

consider the axis-aligned square between (0,0) and (4,4).  we will draw a binary tree inside this box.

the root node is centered on the top edge, at (2,4).  draw an edge from the root to one child at (1,2) and another edge from the root to the other child at (3,2).

next, create two copies of the box so far shrunk linearly by a factor of 2.  place one between (0,0) and (2,2), and the other between (2,0) and (4,2).  the children of the large box become the roots of the small boxes.

repeat recursively, producing an infinite binary tree in finite space.  (this is an example of an iterated function system (IFS).)

inspired by the period-doubling logistic fractal, which requires a lot of computation to produce nice images.  this is easier.

perhaps scale things so that the scale factor is not 2 but the Feigenbaum constant.

instead of straight lines, consider semicircles.  it will look like arches built on arches, a design seen in Roman aqueducts.  or, not semicircles but catenaries.