Realising Equivelar Toroids of Type $$\{4,4\}$$.

By an equivelar toroid we understand a map satisfying the requirements of an abstract polyhedron on the 2-torus where all faces have a fixed number p of edges, and all vertices belong to a fixed number q of edges. We establish that if every regular toroid of type $$\{4,4\}$$ admits a faithful and symmetric realisation in a metric space $$\mathcal {S}$$ then every equivelar toroid of type $$\{4,4\}$$ admits a faithful and symmetric realisation in $$\mathcal {S}$$ . We exemplify this by showing that every equivelar toroid of type $$\{4,4\}$$ admits faithful and symmetric realisations in the 3-sphere and in 3-dimensional projective space. [ABSTRACT FROM AUTHOR]