An unfolding is obtained by cutting along the surface of a polyhedron through all its vetices, and opening all the digedral angles between adjacent faces to obtain a single flat non-overlapping geometric shape.

In this hands-on talk, I will explore connections between these two fascinating concepts, in an attempt to shed some light on the following still unsolved algorithmic problem: How easy(or hard) is it to determine if a given geometric shape can tile the plane?

and the following more artistic and no less fundamental problem: How to create beautiful (or even ugly) tilings?