zvold's blog

Twisted Hexagonal De Bruijn Torus

Hexagonal De Bruijn Torus post describes a few De Bruijn toruses that wrap around on 2 edges, like a rectangle (despite being on a hexagonal grid).

A colleague of mine pointed out that it's possible to wrap a hexagonal grid area on 3 edges, forming a twisted torus. See this page for a nice illustration.

Here's one such shape, found with backtracking:

It has 128 hexagons and, when wrapped in the clockwise way (see below for the explanation), forms a twisted torus containing each of the 7-hexagon neighborhood configurations exactly once.

Note: there are two ways of wrapping the edges, clockwise and counter-clockwise:

                         A A
       A A              A A A
  a a A A A          a a A A
 a a a A A    vs.   a a a . .
  a a . .            a a . . .
     . . .                . .
      . .

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Open questions:

1. Is it possible to find a configuration that would form a twisted De Bruijn torus, independently of the wrapping direction?
2. A large hexagonal area of radius 7 contains 127 hexagons. Is it possible to find a twisted torus containing 127 unique 7-hexagon neighborhoods, omitting exactly one?

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