Abstract
The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions R} (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of R}. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left R}-module R 3 is shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding ternion- induced factorization of the lines of the Fano plane sitting in the middle of the Fano- Snowflake is found to dier fundamentally from the natural one, i.e., from that with respect to the Jacobson radical of the Galois field of two elements.
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