A map on an orientable surface S_{g of genus g is maximally complete if any two faces share an edge and the number of faces is equal to H(g), the Heawood number of the surface. Ringel and Youngs’ work on the Map Color Theorem [11] has shown that such maps exist for all g. However, explicit geometric descriptions of these maps are hard to visualize for g ≥ 3. In this paper, we construct several maximally complete maps on surfaces of small genus in a manner that is visualizable and can be used to produce physical models.}