「Cruciform shapes: in variations -3-」
patterns
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:
- "Alternating homothetic" patterns (in 5 times iterations) -
- A self-similar unit adjusted in scale to compare -
- Corresponding colored diagram in the same scale -
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.

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