Recall that the fractal honeycomb with $27$ outputs arises from blowing up the ten points of the tetractys to larger hexagons.

There are now $81$ internal nodes and $9$ smaller hexagons. Fractal honeycombs nicely keep track of increasing powers of $3$, a basis for $3$-adic numbers. We can either zoom in or zoom out, introducing a process of division by $3$. The central tetractys vertex was marked with the six paths given by the permutations of $XYZ$. These paths are now the separated nodes of a hexagon. In creating this hexagon, we needed a fourth qutrit to form the $81$ paths of a four qutrit word.

8 years ago

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