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

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.

6 pages, 1 table, 1 figure; v2 -- standard representation of the ring of ternions given, 1 figure and 3 references added; v3 -- published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (8)

• Three-Dimensional Gravity Revisited
• Stringy Black Holes and the Geometry of Entanglement
• E_7 and the tripartite entanglement of seven qubits
• Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane
• E_6 and the bipartite entanglement of three qutrits
• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of an Arbitrary Single Qudit
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Cited by in corpus (3)

• Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions
• Introduction to a Quantum Theory over a Galois Field
• A Jacobson Radical Decomposition of the Fano-Snowflake Configuration