From the main c1, we can see that the c1 and c3 cross the vertexes of the pentagons, while the c2 when tangent to the outer pentagon it's also tangent to the c3, crossing the lower vertex of the inner pentagon;
and the c4, shifted to be internally tangent to the star sides, crosses the c3 to the inner poligon vertex and the center of c1, i.e. the system; amazing or not, the two circles [c2, c4] are evenly concentric here: the smaller tangent to the convex pentagon and the other to the concave!
And from here let's drag the c2 circle to cross the center from below, to assist to another stunning effect. This will cross not only two vertexes of the inner pentagon, but evenly the two lower of both the outer ones! Moving down the same to be tangent to the c3, it will again cross the two lower of both the outer, but being tangent to the prolongation of the star vertical sides. Just before ending, it is worth noting that like the triangle APB, and its mirror to PF, ten more golden triangles exalt the pentagon with their multiple facets.
That's all disclosing a beautiful harmony in this kaleidoscope, meaningful once more, thanks to the ‘golden circles’ exibition.
In fact, the time has come [2023], for having matured an ultimate development and deepening in a newly dedicated section.
I guess these are only a few steps, like a start up for further discoveries in the soul of the Celestial Geometry. The next chapter should provide a striking demonstration of that INTELLIGENCE. |