Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 050, 7 pages      arXiv:0803.4436      https://doi.org/10.3842/SIGMA.2008.050

Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions

Metod Saniga a, Hans Havlicek b, Michel Planat c and Petr Pracna d
a) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
b) Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
c) Institut FEMTO-ST/CNRS, MN2S, 32 Avenue de l'Observatoire, F-25044 Besançon Cedex, France
d) J. Heyrovský Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejskova 3, CZ-182 23 Prague 8, Czech Republic

Received May 02, 2008, in final form May 30, 2008; Published online June 04, 2008

Abstract
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.

Key words: geometry over rings; non-commutative ring of order eight; Fano plane.

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