I don't know if I am missing something, but things like trisecting a line or any number division of a line with a straight edge is simple. I assume the same is true of angles or affine space. I just don't feel that it is like magic or even requires some advanced skill.

The only thing that fills me with wonder is that somebody must think it is something difficult to figure out. I surely must be missing some fact about the process.

The line is extended ( parallel ( 2 above ) ,( 1 below ) ) ( bisect lower ) and then diagonal lines are drawn. Enclosed in the circle is the original line.

I read some of the gobbledy goop of alchemists and though they can make astute observations, the problem is too complex to be characterized by a simple relationship and it just sounds like a child explaining nuclear fusion. Plato, Socrates and existentialism is also a load of dung. A person first needs to understand the universe before they can consider their relationship to it.

The image was created with Inkscape using snap to grid with Bézier curves. Elapsed time 1 minute. There are some new options to Inkscape.

I will have to investigate the math a bit to see if this holds on transformation and rotation.

I would think so at first glance, at least in Euclidian space. UNSW Linear Algebra.

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