Greg Egan on Nostr: These are the 6 vertices whose hexagon centre is the identity matrix, multiplied by ...

These are the 6 vertices whose hexagon centre is the identity matrix, multiplied by √6 to give Eisenstein-integer matrices.

The dot product between a hexagon centre and any of its vertices is –√(3/2).

The dot product between adjacent vertices is –5/4.

The dot product between nearest-neighbour hexagon centres is –3/2.

These particular 6 vertices lie on the two horospheres defined by dot products of –√(3/2) with the two null vectors (1,0,0,±1).

Empirically, all the vertices in the honeycomb seem to be matrices with Eisenstein integer entries divided by √6. The diagonal entries are always multiples of 3/√6.