http://wp.me/p1fcz8-4×2 got us as far as understanding that from the idea that a certain number of wires must cross when fabricating an enigma wheel one can move beyond this 1d wiring crossover world to n-dimensional “wire crossover worlds”. While this can be thought of as a compound wheel set (eg 2 tunny wheels), such is really still a 1d world (now with longer period). We really want to capture true n-d wheels.
Expander graphs seem to give us what we need. To each dimension of the vector space we see that an associated eigenvector aligns with the nth eigenvalue of the graph’s adjacency operator. As in the tunny writeup (1945), one can cast the analysis in terms of a fourier basis enabling one to turn to the weight classes as ways to sort values, since – in this basis – one can take the inner product of the each unit eigenvector (in the standard basis) and the generator function.
We eventually get to see a concentric set of circles (or spherical orbits within orbits, said better) in which are located certain subsets of graph nodes. Between the shells are some edges, that play the role of wiring crossover cables (between shells, or weight classes). And this number we may want to count.
Now we have mental model of a true n-dimensional wheel. The wiring metaphor only reappears in the “convolution space” of 2 wheels whose zigzag product gives a compound wheel with the required space, that is an expression of the weight classes (being related to the standard basis, or combinations thereof). And we are concerned now with the crossover edges between layers (when in phase/convolution space), rather than literal wiring plan crossovers (in literal space).
Now it makes more sense to go back to molina’s phd, recalling how she focussed on a group whose terms were crossover sets and whose binary operator wired together such crossovers.
It makes alot more sense niw to consider the inner product of the function and its lagrangian since this looks at the crossover specific to a particular inter layer channel, being an edge focussed measure (derived from differencing the nodes at each end of the edgez in distinct layers of course).