Yawn! Only five hours later, curiosity lured me out of bed: 
How did Convincing Patterns emerge from Architectonic Points, the design I was admiring "on my way" to bed?
The answer was obvious this morning:
The center scalene triangle (golden with a 3-D appearance), was the key that unlocked this design's geometric potential: I knew that the geometry of the triangle was symmetric with the larger and smaller scalene triangles. And I knew that its size was proportional to this geometric pattern of nested scalene triangles.
Long story short: I visualized* how the design (the three scalene triangles would look if positioned similarly on the plane. The mental image shouted: "Go for it! Forget about bed - another sunrise bedtime is only that (another sunrise bedtime)." Once this geometric key was turned (effectively counterclockwise), the expanded geometry became obvious - just connect the points with straight lines, and select colors from the pallet already in use.Imagine!
This geometric key is also evidence of a direct relationship of the square root of Pi (the diagonal of that square) and the square root of 2 (the circle's diameter.* Visualization of moving geometric objects became an acquired, critical skill in last year's geometry exploration.
Rod ...
... 
Re: Convincing Patterns of Pi (design updated)
The Convincing Patterns geometry might reveal an "esoteric singularity" 
When the side length of the square of a circle is attached at either end by the side length of the circle's inscribed square, and the two attached sides are rotated clockwise until their opposite ends may be connected by a vertical straight line, the circle and its square are effectively united and precisely defined. The length of the vertical line is then equal to the length of the side of the circle's square. 
In the Convincing Patterns design, the leftmost green diagonal line (left side of the largest green scalene triangle), the lower horizontal green line (horizontal side of this scalene triangle), and the lower golden diagonal line comprise one set of three lines in this squared circle singularity.
It's already past 2:00 AM and, because an esoteric singularity is a "tough act to follow", I was going to skip this next comment. But a glance at the 2:22 AM prompt persuaded me to reconsider:
).
When we're proposing a "singularity" of multi-century circle squaring, a disclaimer often keeps us on the preferred side of the sanitas/morbus border. So, today's disclaimer is that this singularity is still in development (IMO) - the geometry shows what we might expect when a circle is squared (or three nested, as in this design).
Also, a correction in visualization: the lower golden line of the set is moving counter-clockwise - not clockwise. And this "movement" indicates that the red square(s) moves from upper left to lower right at 315 degrees; the length of the vertical line connecting the points of the rotating lines is always equal to the length of the horizontal line in the set.
So, I opened a third application (screenprint) and copied the screen clutter.
"What's with the square red plates?" the first guest complained. "And where's the rest of my entrée?"
"Your deed is certain ... not impossible - just difficult."
Ditto is a complex sample of the seemingly infinite number of patterns that can be found in squared circle geometry. Once belief is entertained that a circle can be squared, these many patterns confirm that belief and suggest geometric pathways for discovery of the "impossible".