Are they the only choices?  Perhaps there are a series of hexagons and or pentagons such as what a football / soccer ball has which forms the basis of the planet's crust?

These areas (we'll refer to as plates) could then be semi anchored to each-other with an area of free movement where the plates can rub together at the edges of the plates.

Care should be given if living in these areas as they may over time develop into a "ring of fire" if the plates continue to rub together.