M Theory Lesson 149

Now the hexagon that runs through the three discs of the pants in lesson 144, and along the real axis on the Riemann sphere, is just like the hexagon (cyclohedron) from yesterday, because three edges are labelled by single chord hexagons (the squares of the associahedron in 3 dimensions) and the other three edges by edges in the associahedron which link the squares. The two vertices of the associahedron which do not appear in the circuit correspond to the two vertices of the trivalent trees drawn on the pair of pants.

So this hexagon is a real dimension shifter! Previously, the three squares were associated with three faces of the mass generation cube. The completion of the cube is now seen as a pairing between the two triangular circuits of the hexagon, denoted respectively by 1-circulant and 2-circulant matrices in the Fourier transform.