Good points, Carl. Yes, I can’t help thinking the zero angle circulants are natural basis elements for something…. see this old reference.

]]>The square of a 2-circulant is a restricted type of 1-circulant, the off diagonal elements are all equal. In the Koide formulas, that would be a system with a “that damned angle” of zero.

Interestingly, if you attribute “that damned angle” to soft bosons, then the electrons do have an angle of zero. They would be consistent with being the square of a 2-circulant.

But the neutrinos do have a geometric angle, pi/12.

]]>I do not have any understanding of this type of notation, so I have two dumb questions for possible “alternative perspectives”.

1 – When I first saw your diagram, I was thinking that perhaps the black nodes represented real and the open nodes “imaginary” dimensions in complex space?

2 – Although the direction of circulation is probably clockwise by definition, is a counterclockwise circulation possible?

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