Twin “Fano-Snowflakes” over the Smallest Ring of Ternions

Given a finite associative ring with unity, $R$, any free (left) cyclic submodule (FCS) generated by a unimodular $(n+1)$-tuple of elements of $R$ represents a point of the $n$-dimensional projective space over $R$. Suppose that $R$ also features FCSs generated by $(n+1)$-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for $n=2$ when $R$ is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a “Fano-Snowflake” due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations – each being tied to either of the two maximal ideals of the ring – which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined.