Behind the Intuition of Tilings

It may seem visually intuitive that certain sets of tiles can be used to cover the entire plane without gaps or overlaps. However, it is often much more challenging to prove such statements rigorously. The Extension Theorem justifies the visual intuition. It allows us to prove the existence of a tiling by covering a circle of arbitrarily large finite radius. We clarify the proof of the Extension Theorem, consider its necessary assumptions, and present some interesting generalizations.