ManyFolds in another Variety

The basic self-similar (sub) structure of the iterative [almost self-] similar flatly foldable pattern appeared in the previous article is:



The above polygonal structure has eight edges (colored in red) as its boundary: these edges come from two - large and small - squares that are placed concentric at 45°[π/4 (rad)] each other.
The (homothetic) ratio of the large square to the small square is
(√2 + 1)/1 [= 1/(√2 - 1)].

Such closed and connected regions, each of these is homeomorphic to a 2-dimensional closed annulus [/compact cylindrical surface], will make "alternating homothetic" [this "expression" might not be correct in the strict sense of mathematics...] patterns over a square as follows:




Click here to see a larger image - in 1285 pxls by 1285 pxls - of the above diagram; and here to see its corresponding patterns by the regions over a square in the same scales.