Aperiodic tiling, tessellation, and crystallography
Stoyan Smoukov and H. K. D. H. Bhadeshia
The image below is of tiling done by Stoyan Smoukov (who kindly provided the image) at a sink in his house.
The tiles have a clever shape so that they can tessellate to fill completely, a two-dimensional surface. Each tile is identical in shape, rather like a shirt with a neck and two sleeves, but can have different orientations relative to the axes of the wall, and can have a different colour. However, they clearly can be rotated relative to those axes in order to fill space.
The two tiles marked with red squares have exactly the same colour and orientation, but do not represent lattice points because their environments are different, as is evident from the tiles marked with red crosses (these two tiles may be in the same orientation but have different colours).
There is, therefore, no long-range periodicity, so the pattern does not qualify as a crystal.
In summary, we have discussed a unique tiling pattern utilised in a domestic setting that was discovered originally by David Smith. Known as the "hat" tile, this specific shape is capable of covering a flat surface entirely without leaving any gaps. Although the individual tiles share an identical geometry, they are arranged in various orientations and colours across the wall. The author explains that the arrangement lacks long-range periodicity, meaning it does not repeat in a predictable, crystalline manner. Because the surrounding environments of identical tiles differ, the pattern is technically classified as non-crystalline rather than a standard lattice. Consequently, this mathematical curiosity demonstrates how a single shape can tessellate while avoiding traditional symmetry.
