Can this shape tile the plane? It is not easy to guess by just putting the shapes together. With a little thought however there is an easy proof. It can't. How far can it go though? How many rings of tiles can you build around a first one. This is called the Heesch number of the tiling.
Cut out of acrylic/card/wood/stone using your favourite cutter, laser water-jet, craft etc. Cut file set up for A4 sheet.