M Theory Lesson 211

Generalising the matrix $K$ further still by adding phases to $\left(2\right)$, one can offset the exact Koide eigenvalues for a phase $\theta =\frac{2}{9}+\frac{2\pi n}{3}$ in the component $\left(1\right)$ by a very small phase $\delta \simeq 10^{-7}$. Note that the cyclic $\frac{2\pi n}{3}$ factors from the Fourier transform have not yet been accounted for in the basic $6\times 6$ eigenvalue problem, so they are included in $\theta$. The charged lepton $K$ operator eigenvalues for the democratic 6-vector then take the form

$\lambda =a+b\text{cos}\theta +c\text{sin}\delta$

where the last term is considered a small electric field term. That is, by making the amplitude $c$ small, $\delta$ need not be small and one could take $\delta =\theta$. This indicates a correspondence between, on the one hand $\left(1\right)$ and magnetic fields, and on the other $\left(2\right)$ and electric fields, which holds even if the factor $\left(2\right)$ becomes a 2-circulant matrix. Electric magnetic duality then swaps the two triangles making up a basic $S_3$ hexagon, as previously discussed. For the cube this may be viewed as the duality of a triangle and trivalent vertex in the plane. That is, a duality for Space and Time.