This talk is about a tiling problem. I will present some preliminary results of a joint work with Heidi Burgiel. In this work, we are attempting to extend some recent results of Wang and Deguang. About fourteen years ago, Wang and Deguang proved that two lattices of the same volume have a common fundamental domain. However, Wang and Deguang did not address the topological properties of these common fundamental domains. For example, nothing was ever said about connectivity. In our work, we are interested in characterizing pairs of lattices of the same volume which have connected fundamental domains. I shall discuss our recent progress, the techniques used in our approach, and some of the serious challenges that we are facing. The content of the presentation is appropriate for undergraduate math majors. So, please encourage your students to attend.