Firstly, a correction. Previously, with 'old' site, on the Penrose section I showed a series of tessellations of kites and darts (and thin and thick rhombs) with bird and fish motifs of which I believed showed nonperiodic tilings. However, subsequently, I have found that I was mistaken, in that these could be placed as a periodic arrangement rather than non-periodic as was thought. I now correct this error, by showing how it is possible to outline a par hexagon (in red), and as this unit tiles by translation, that unit will tile periodically, and so the ‘unique’ aspect of the Penrose tilings (nonperiodicity) is lost.

The cause of this error is the introduction of a line of 180° rotational symmetry, of which such a line can be considered as inadmissible. Instead, the lines must consist of asymmetric lines. For tuition purposes, namely of what to avoid, I have decided to retain the diagram ‘as was’, and outline a par hexagon. If the diagram is examined closely, it will be seen that for both long and short sides the lines consist of 180° rotation. What about if the tiling has just one line of rotation, will it tile nonperiodically? The answer is no - even just one line possessing 180° rotational symmetry is inadmissible, as a par hexagon can still be outlined.

In a sense, this is somewhat unfortunate situation, as a line of 180° symmetry is ideal for both bird and fish motifs, as shown, with the motifs of high quality. Therefore, this thus causes a workaround to be put in place, in which the line retains aspects of this, but has 'faux' 180° symmetry (not shown).