Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions

TL;DR

This paper explores a novel geometric structure called 'Fano-Snowflake' arising from non-unimodular free cyclic submodules over a specific non-commutative ring of order eight, revealing twin configurations related to quantum and black hole theories.

Contribution

It introduces the concept of Fano-Snowflakes as geometric structures from non-unimodular submodules over a unique non-commutative ring, expanding the understanding of projective geometries over rings.

Findings

Identifies two twin Fano-Snowflake configurations tied to the ring's maximal ideals.

Shows the Fano plane appears as a core component within these configurations.

Suggests potential applications in quantum information and black hole physics.

Abstract

Given a finite associative ring with unity, $R$, any free (left) cyclic submodule (FCS) generated by a $uni$modular ($n+1$)-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.