The tetractys for three trits came from the commutative projection of the $27$ noncommutative paths. We can place all $27$ paths on a three dimensional torus $T^3$.

The front face contains paths starting with $X$ and the rear face paths starting with $Z$. Everything is cyclic, so opposite faces are glued together by extra edges. Now the vertical diagonal faces (colour coded) correspond to the trefoil quandle $B_3$ generators $a$, $b$ and $c$. Ignoring the words $XXX$, $YYY$ and $ZZZ$, each such face has eight vertices on an octagon. There are always two neutrino words of type $XYZ$, which we take to be the source and target of a cube. Each colour could encode a Fano plane, with a trit word for each octonion unit in the plane.

However, we will have to play more with the quandle rules to find the optimal choice of quaternion lines for the Fano plane. We would like to create Fano planes from the hexagon on the commutative tetractys. So there is a correspondence between the $27$ paths and the dimensions of the $3 \times 3$ exceptional octonion Jordan algebra. Let us check how quandle rules might express the quaternion rules, along one line in the Fano plane. Define a two word product $(123)(456) = ((14)(25)(36))$. Use the quandle rules $XX = X$ and $XY = Z$ etc. Then we see that $(YXY)(XXZ) = (ZXX)$, as required. Each unit is given by a word of length three. A child could understand quantum gravity.

It from bit and trit. Space from knots. Algebra from knots.

8 years ago

So far we have not complexified the algebra. If we add left and right brackets to the length three trit words, there are now $54$ paths: the dimension of the bioctonion algebra. Thus a possible interpretation of Koide phases is: a phase is a mixture of left and right brackets.

ReplyDeleteSome of my ex mathematical colleagues are quite fond of tori and Fourier transforms and nonassociative algebras. Ah yes, the Fourier transform: as usual, we like the $3 \times 3$ Fourier transform, which acts on a three point space. As previously noted, a twisted Fourier transform sends fermions to bosons. The twisting adds braid information to diagrams.

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