Sunday, April 20, 2008

Fano Matroid is Spot-on

In this blog I give reasons for choosing a rendition of the fano matroid as the logo for Spot-on Solution, as shown in this image.

The fano matroid is named after Gino Fano, a mathematician who devoted his life to projective and algebraic geometry. The fano matroid is also referred to as the fano plane, projective geometry P(2,2), steiner tipple system S(2,3,7).

During my foundation in operations research at the University of Montana, I was introduced to all three of the notions associated with the algebraic structure comprising 7 vertices and seven edges. When a cube with 8 vertices and 8 edges (3 dimensional object) is projected on to a plane (2 dimensional space) it produces the fano plane. In block design theory the steiner triple system S(2,3,7) when represented as a graph, resembles the fano plane. In matroid theory the fano matroid plays a significant role as what is termed as an excluded minor in differentiating properties of certain classes of matroids.

My masters thesis was on defining the notion of circuit double covers for matroids and developing a few algorithms to create matroids with circuit double covers in several classes of matroids such as the uniform matroid, graphic matroids, ternary matroids, binary matroid, etc. The notion of circuit double covers had already been established for graphs in graphs theory by Seymour and Szekeres, in 1970. This conjecture, the cycle double covers for graphs, remains an open problem to this day. I first came across the notion of cycle double covers of graphs when attending the 1998 Bigsky conference on Discrete Mathematics, hosted by the University of Montana Mathematics Department. My adviser, Prof. Jenny McNulty, handed me the challenge of understanding circuit double covers of matroids, which was a pioneering topic. That's when I fell in love with the fano matroid, which is now my favorite abstract structure, where 23 is my favorite number.

With this I have chosen a rendition of the fano matroid as my Spot On Solution's logo. As seen in the image above, the warped version is due to many reasons. Given that my current focus on systems, specifically, Information Communication Systems (ICT), the idea of stability is a major criteria that all those who deal with systems hard to establish. Although perfection cannot be achieved, designers of systems, strive to achieve, at least, an asymptotic or near perfection, of stability. The lines (or edges of the fano matroid) in the logo resemble the transient and steady state response of a system arriving at some asymptotic stability (looking like an exponential function). Moreover, no matter at which "state" (vertex on the fano matroid) one starts it is possible to traverse to any other point with one hop (passing through one vertex).

In ICT redundancy is a word that is commonly used as well as an action that is commonly practiced. A double covering in graph theory or matroid theory is synonymous with redundancy. Remarkably the fano matroid is beyond a double covering and is a triple covering per say; where each node has 3 edges descending upon it. With respect to ICT we can think of it as each node is connected to 3 independent channels or links; thus disruption of one link does not effect the operation of any of the nodes; as a matter of fact each node can afford to lose 2 links and still be able to communicate with the other nodes.

Credit has to be given to Absoulte Einsteins for creating the first cut of the logo and Dacia Closson, my friend with exceptional graphic and web design capabilities, for completing the logo.