M Theory Lesson 353

On the surface across the corner of the cube we can draw the points of the path space up to permutations. There are nine points around the triangle and one central point representing $XYZ$. Let us include an edge whenever a trit is flipped. If an outer edge is weighted by $1/2$ and an inner edge by $1$ we recover a Pythagorean tetractys labeled by both path types and weighted valencies at a node, the latter corresponding to the path count on the cube. This is a more cyclic picture than the use of projective degree that we saw for the associahedron ternary triangle, reflecting the need to replace duality by triality.