M Theory Lesson 111

A commenter at Carl Brannen's blog has noted the similarity between the snuark mass computations and the Fano plane, which we recall describes the octonions via a cube with corners $1,e_1,e_2,\cdots ,e_7$. In Carl's notation, the correspondence is

$e_1=+y$
$e_2=+z$
$e_3=-y$
$e_4=+x$
$e_5=-x$
$e_6=-z$
$e_7=0$

where $0$ is the fictitious vacuum and the $1$ is hidden at the rear of his diagrams. On the octonion cube, this gives a source of $0$ and a target of $1$, with $x$, $y$ and $z$ axes running through the other three full diagonals. Note that a projection of this cube onto a plane results in a hexagon with vertices $+x,-z,+y,-x,+z,-y$, which has a selected triple of nodes $(x,y,z)$ as previously noted. The basic simplex formed from the source in these directions is the area marked with the phase $\frac{\pi }{24}$ in Carl's computation.

Aside: Note also the new paper by Yidun Wan on 3-braids, which discusses Veneziano bubbles and rotations by $\frac{\pi }{3}$ and $\frac{2\pi }{3}$, the symmetries of a triangle.