Greg Egan on Nostr: Here’s a construction of a portion of the honeycomb that I think can be extended ...

Here’s a construction of a portion of the honeycomb that I think can be extended indefinitely.

The black point at the centre corresponds to the identity matrix I.

The 6 cyan and 6 yellow points are the 12 nearest neighbours of I in 𝔥₂(𝔼):

\[\left(
\begin{array}{cc}
2 & (-\omega )^k \\
(-\omega )^{6-k} & 1 \\
\end{array}
\right)\]

and

\[\left(
\begin{array}{cc}
1 & (-\omega )^k \\
(-\omega )^{6-k} & 2 \\
\end{array}
\right)\]

where k=0,...,5.

Grey lines join all nearest neighbours among these 13 points.

The 6 blue vertices that form a hexagon centred on the black point are all equidistant from the black point, one magenta point and one yellow point. They are:

\[\frac{1}{\sqrt{6}}\left(
\begin{array}{cc}
3 & (1-\omega ) (-\omega )^k \\
(1-\omega ) (-\omega )^{5-k} & 3 \\
\end{array}
\right)\]

We then map that central hexagon to 12 others, using Lorentz transformations that take I to each of its 12 nearest neighbours in 𝔥₂(𝔼), and that take one hexagon edge to the opposite edge.

Using the action:

A ↦ gAg*

the 12 matrices we use for g are:

\[\left(
\begin{array}{cc}
1 & (-\omega )^k \\
0 & 1 \\
\end{array}
\right)\]

and

\[\left(
\begin{array}{cc}
1 & 0 \\
(-\omega )^{6-k} & 1 \\
\end{array}
\right)\]