The Cartwheel Tilings

To create the cartwheel tilings, start with the ace.

        Then inflate repeatedly.  Recall that inflation is a decomposition followed by an enlargement,
in this case by a factor of the golden ratio.  An illustration, without the enlargement, i.e. a decomposition.

 

In each of these tilings, there is only one symmetry, namely a reflection through a vertical axis.  With a little trimming, C_{2 ,}  
C₄ , C₆ , have 5-fold rotational symmetry as well if the whole patch, rather than the individual tiles is considered.

Note also that the odd patches are upside-down in the even cartwheels.
The even patch C_2n is called a Cartwheel of order n.
By the Extension Theorem, there is a cartwheel tiling of the plane that contains each smaller cartwheel in a concentric fashion.

Cartwheels Part 2  Aperiodic Tiling