I’m not sure why, but mostly I’ve been finding tesselations based on what I call h-lines and v-lines. (I don’t know the (or if there is a) standard term.)

They look like this:

And lately sometimes the square lengths are 0 or negative; this can be handled by this system too, and looks like: (I put a circle on the corner where the 0 is, to remind me where it is.)

So. I’ve found two equivalent methods for constructing these, the first by writing arithmetic series on am h-line/v-line grid, the other by drawing 2 rows of parabolas of squares, one row at 90 to the other.

First method

Start with a single h-line. e.g. 3 4

5 2.

Then extend the 2 diagonal arithmetic series (they have the same common difference). Next select another arithmetic series to alternate with that one, i.e. write a number to the right of the initial 4. This will determine the entire tiling. Fill in the entire grid. There are two alternating arithmetic series going diagonally SW-NE, and 2 going SE-NW.

Then drawing the tiling is fairly simple, although the negative numbers and 0 can be tricky, so do those areas last.

Also, I haven’t studied this in detail.. It seems doing it this way, that the tiling is actually of a ‘flattened’ Riemann surface, the one for , like a double pancake that crosses over itself at a seam. To be a plane tiling, only one of these halves is drawn. I’ve drawn some where the seam isnt flat.

I’ve noticed that no matter where the 0s are in the grid, (often but not always in an X pattern) they are always in a straight line in the tesselation.