OS PRIMOS DISCRETOS DO PLANO AFIM

Abstract

Finite geometries form a fundamental link between classical geometry, abstract algebra, and combinatorics. This outreach article attempts to capture a brief portrait of this multidisciplinary character of certain geometric structures on finite sets. Combining an axiomatic approach with combinatorial arguments, we discuss incidence and parallelism conditions that characterize finite affine planes, the ``hermit cousins'' of the acclaimed Euclidean affine plane. We present the construction of such planes from Galois fields. In addition to their mathematical importance, it is worth mentioning that finite geometries emerge in contemporary applications such as error-correcting codes theory, statistical experiment design, as well as in recreational games.

Keywords: finite geometry, affine plane, axiom independence, bijections, modular arithmetic.