Linear elasticity theory predicts a divergent strain field at the dislocation core, resulting from the continuum approximation breaking down at the atomic scale. We introduce a minimum model that includes elastic interactions and discrete lattice periodicity, and derive a set of equations that treat the core of an edge dislocation from a solely geometric perspective. We find an analytical formula for the displacement field of a straight dislocation of arbitrary mixed character, and we predict that the dislocation core widens as the screw character becomes more dominant. This finding is in qualitative and quantitative agreement with atomistic simulations of mixed dislocations in tungsten. The theory is based on a continuum form of the multistring Frenkel-Kontorova model, which is a nearest-neighbor model for atomic bonding that also takes into account the discreteness of the crystal lattice. Thus, we circumvent the need to use adjustable parameters in the treatment of a dislocation core.