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.
Jan 2017: Ed Pegg gave me the great idea of using complex numbers for rectangles! So trying the same thing but with rectangles… with (obviously) for width, for height. The real components will have to agree along h-lines, the imaginary along v-lines.
Start ‘as before’ with extending an h-line:
But now an ‘h-line’ really is a line – there’s nothing to draw yet; we’ve only added widths. Even after adding an arbitrary diagonal of heights/imaginary numbers..
..there’s still nothing much to draw. So let’s add two more diagonals – an integer one through the , and a totally arbitrary imaginary one in the same squares..uh, rectangles.