asks the question, what is a quantum honeycomb
? Recall that the usual
honeycombs for 3x3 complex matrices involve a single hexagon in the plane. Let us resolve the vertices of this hexagon into hexagons.
M theory requires octonion Hermitean matrices. Instead of a combination of three matrices resulting in another complex matrix, with 9 external edges, we consider 27 external edges in the triple strand case. Note the scale invariance of the diagram in zooming either inwards or
outwards. Altering the local geometry does not affect the rigidity of the pattern. Apologies for the omission of the top Y from the diagram and for the wonky edges.
Drawing a triangle around the triple strand diagram with a vertex at the top of the diagram, we observe that outside the central hexagon there will be 6 pentagons and 3 squares. Thus the cylinder between the triangle and the hexagon happens to have a tiling by the faces of a Stasheff polytope. We could glue two such cylinders together to make a pretty torus geometry.