M Theory Lesson 354

Taking noncommutative paths of length $d$ in $n$ variables to their endpoint monomials results in a planar complex of measured simplices. The standard simplices form one column, as shown in the table. If we replace $X$ by a zero and $Y$ by a $1$, the row of measured intervals condenses the $d$ qubit objects. The secondary polytope in this row (see the green book) is the cube. For example, the interval of length $3$ maps to the square, because the triangulations of this simplex are given by the partitions of $3$, namely the partitions $(1,1,1)$, $(2,1)$, $(1,2)$ and $(3)$. Moreover, this square secondary polytope also comes from the noncommutative paths for the interval of length two, since we can draw Young diagrams around the nodes of the planar lattice of noncommutative paths. In higher rows, these standard partition maps may still be applied to the edges of a higher dimensional simplex.

The first instance of a simplex with central vertices is the tetractys. This simplex came from the $27$ length $3$ paths on a cubic lattice. The central vertex stands for the six paths in three distinct variables, namely the six permutations in $S_3$. This is the first vertex weight that is the product of two prime numbers. Volumes of pieces in a triangulation (of a simplex) are also naturally expressed in terms of prime factors, once a minimal volume is normalised to $1$. Such volumes determine coefficients for generalised entropy measures.