It is likely that there are several order 5 tiles in addition to the order 5 dissections of the 8 order 3 tiles and the additional 58 order 5 tiles that can be derived from the order 3 tiles. However order 5 tiles are harder to find than order 3 tiles and I have only identified a single such tile.
An example IFS for this tile is { p → -ap + 1; p → -a2p; p → a2p + 1; p → -a3p + 1; p → a4p + 1 }
There is an alternative dissection - the right hand 4 elements can be rotated by 180° around their common centre.
It can be seen that those 4 elements form a symmetric tile, and hence that this tile has a periodic tiling of the plane with unit cell signature 0123. The unit cell transforms are { p → p; p → -ap; p → a2p; p → -a3p }.
© 2017 Stewart R. Hinsley