They’re built with alternating (what I call) h-lines and v-lines. An h-line consists of four squares, two on either side of a horizontal line, with equal sums and aligned edges. The tesselation is first written in digits on a square grid marked with lines like a wire screen door, then constructed by drawing any h- or v-line from the grid and working outwards from there.

Start with any h-line: e.g. :

Extend the arithmetic progressions along the diagonals:

Then fill in a diagonal next to one of these 2, with any arithmetic progression. e.g:

All the empty squares have now been determined and can be filled in, making the sums equal across all line segments (the h-lines and v-lines).

Then when drawing the tesselation, draw the squares of negative size on the other side of the line from where they ‘should’ be, e.g. :

It seems that any grid of numbers constructed this way makes a tesselation of the plane.

Also, it seems that all the -sized squares (I mark them with a small circle) lie in a straight line in the tesselation, no matter what sequences are used. I’m sure there’s an obvious reason for that!

The square packings I found a few years ago are each a subset area of one of these tesselations, where there’s only 1 of each sized square.

[Dec 2016]