exupero's blog

Semi-tessellation with a continuous line

Another product of long meetings, I found it's possible to tessellate plus symbols:

What I didn't know until recently is that this can (almost) be drawn with a single continuous line that only self-intersects at corners:

Drawing it this way hits a dead end, though. When the line returns to the point where it started, it has no escape route. We'll have to reverse direction around the central plus, switching from counter-clockwise to clockwise, and we'll have to do it before the next plus would land us on our starting point:

Further, but apparently we'll have to double back every time.

Carrying this approach to several more iterations produces

It seems if we don't want to run into a dead end we're stuck with a gap in the pattern and we can't fully tessellate the plane.

This dilemma isn't unique to plus-shaped tessellation. Squares exhibit the same limitation:

We can, however, fill in the gap once we reach a desired size:

which also works for plus-tessellation: