The kite and dart can be found in the pentagram. If they are allowed to be put together like this, then they tile in a periodic fashion. There are various strategies to prevent this. Some involve color-matching schemes on vertices. Others use decorations on the tiles, and the decorations have to match up. Still others involve alterations to the shape of the tiles, as below.
The golden ratio in the pentagram
If the various matching schemes are followed, or if the tiles are altered as above, then the kite and dart are an aperiodic set, i.e. they will only tile the plane in an aperiodic fashion. Plenty of tiles, e.g. squares, can be made to tile the plane in a nonperiodic fashion, but since a square will also tile periodically, it is not an aperiodic tile. A tiling by kites and darts.
