M Theory Lesson 211

Generalising the matrix $K$ further still by adding phases to $(2)$, one can offset the exact Koide eigenvalues for a phase $\theta = \frac{2}{9} + \frac{2 \pi n}{3}$ in the component $(1)$ 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

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

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 $(1)$ and magnetic fields, and on the other $(2)$ and electric fields, which holds even if the factor $(2)$ 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.