We present a set of DNA supramolecular architectures based on the polymerization of discrete DNA tiles having the shape of parallelograms and designed to have a one-dimensional inter-tile connectivity. Tiles bind to each other with two connections, which have different thermal stabilities. We discuss how this difference in stability implies that the same monomeric tile can yield supramolecular polymers of different shapes just by changing the polymerization conditions. We show how this system reacts to external stimuli by interconverting between some of its possible states. Concurrently, we show how performing the polymerization on a surface can influence its outcome.