In Chains (Annuli I)

A cube can be sliced half so that the cross section is a regular hexagon, and this even in four different ways.

Hexacut

In particular, we can place regular hexagons into space so that the corners all have integer coordinates, and the hexagons face four different directions. This suggests to interlink the hexagons, for instance like so:

Minihex

The mathematician immediately will ask to put as many of such linked hexagons into space, and this automatically drives the discovery of new structures or leads to connections with the already known.

In this case, it turns out that feeding one hexagon through the center of another is not so smart.

Smallhex

When the center spot is taken, any further hexagon through either of the first two must be placed by breaking the symmetry. Above we have threaded three hexagons through a horizontal red hexagons in a rotationally symmetric way, and placed a mirror image of this tetrad below. While these two pieces are not yet interlinked, they can together be translated as to create a very tight interlinked system of annuli (replacing the hexagons with smooth annuli).

Hexasphinx1

This works in all directions, and being stuck somewhere within this tangle will look like this:

Hexasphinx2

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