On reading how the zigzag graph works, I found myself applying the ideas to turing’s use of the fano plane.
The point to observe is that the fano plane is a set of design blocks: with the 3 point of each line logically affiliated with an opposite point (on the unit circle). One should think of each as a cloud, having a particular conditional distribution with respect the graph as a whole. It’s the zig (or the zag).
Now the point about zigzag is that the semi direct product generates an averaging process that diminish certain vector lengths – those vectors that are, specifically orthogonal to the constant functions in the space. The proportion of diminishing is a function of the spectral gap of the character function.
The nearest neighbors are clouds, that is. Or they ‘re the set of blocks, equivalently.
So what is the hyphen function of zigzag, in the fano plane?
It’s the unit circle, and its permutation centric automorphisms acting as a transducer of entropy induced by the zig, or an entropy generator should the zig have contributed none.
In geometric terms, folks are reducing the angle between the constant function and those vectors in that special space to which the averaging process uniquely applies. What the zag does is renormalize and recentralize that space (so it’s orthogonal again, in the new norm now) ready for the next round.
Thus you see how the conditional probabilities are transformed.
For the first time I think we understand the 1950s response to the attacks mounted against tunny and purple.