In searching for a good geometric system for the tiling of the pillows, we have become very interested in pentagon tiling. This system allows for a great deal of variation in the pattern while using the same shape. While we are not entirely constrained to use only one shape, this will make the fabrication process easier, and allow us to continue to design the overall shape after we start production of the individual pillows.
The definition of pentagon tiling patterns is a rich mathematical subject, with new patterns continuing to be discovered. So far, 15 have been discovered, the most recent in 2015. We are focusing on types 7 and 8, as they produce the most interesting, non-repetitive patterns:
Good sites as a reference: