i.e. Is there a specific reason why these diagrams exist? Is the "neatness" of the design simply a graphical representation of the mathematical constants, or is there some underlying reason why a person would envision this?
To be clear: I am asking if there is a physical phenomenon being portrayed or is this just a mathematical trick that is pleasing to the eye?
I have seen several depictions and they vary considerably in their complexity. Example: https://www.youtube.com/watch?v=f7o0kah6soA
Here is a site that is less of an incoherent jumble of science jargon. (although it still says nothing about the underlying reason for applying it) http://weise.symmetry-us.com/pythagorean-approach-in-chemistry-tetrahedron/