That a triangle can be divided into 16 similar triangles is not so mysterious (to geometers),Mysterious pattern of "Sweet 16" in circle-squaring right triangle

that can be divided into 16 similar triangles, where 16 = 1 + 3 + 5 + 7

but how does 16 (instead of other increments) relate to this squared circle geometry

Bubble eQ16 shows a meaningful relationship with further emphasis on sqrt(2).

An intriguing geometric clue (re: 16 / 2 = 8 ) ...

A circle and its square connect at 8 symmetric points ... only

Rod