making tiling patterns with the new aperiodic monotile
using David Smith's "hat" tile to make Escher-like patterns
I read about the recent discovery of a tile shape that is the long sought “Einstein” block: a single tile that can tessellate (completely fill a flat plane without gaps). It also never repeats the pattern in a periodic fashion so never forms a 2D lattice. Pictures of the tile and its tessellations were published in New Scientist (1 april 23), in the Guardian and Times newspapers (4 april). The small fragment of the pattern shown in the New Scientist suggested a three-fold rotational symmetry, so I wondered if this could be used to generate such a tessellating pattern with the hat shaped block, the so called “Einstein”.
Playing with this tile it is easy to see that a particular arrangement of three tiles fit together around a hole which is the shape of the mirror image of the tile, so is filled by the tile flipped over. The resulting 4 tile triangle can combine with two other copies in a unique way about a hole shaped like the original tile to form a 13 tile large triangle.
The resulting 4 and 13 tile triangles can combine, separated by single rows of the original hat tiles, to make larger patterns. Choosing different ratios of the 4 tile and 13 tile triangles give different possible patterns, some of which tessellate as claimed, and some which do not.
For the 1:1 ratio, imposing a three-fold rotational symmetry, so the whole pattern superimposes on itself when rotated by 120 deg, gives a particularly pleasing pattern reminiscent of an Escher etching
.The three fold axis of symmetry perpendicular to the page, passes through the center of the hexagon. The pattern looks very ordered, with perhaps a unit cell containing just four of the 4 tile triangles, but close examination shows it never repeats, the hat rotations mean the smallest unit cell of the pattern is the whole pattern itself. This pattern does not tessellate the plane. It has gaps (the 6 black double diamond shapes) which sum up to an area equal to 1.5 tile areas. Extending the pattern further out, finds an increasing number of such gaps.
If the ratio of 4 tile triangles, to 13 tile triangles, is increased to between 2:1 or 3:1, patterns can be found which do tessellate and are non-periodic as previously reported in the newspapers where they are depicted
This raises the question “is the failure of the 1:1 ratio to tessellate due to the imposed threefold symmetry or will all 1:1 patterns refuse to tessellate, and if so why?” .
I think all 1:1 ratio patterns where the triangles are separated by single rows of tiles cannot tessellate, gaps are inevitable. This can be seen by looking at the edge patterns of the component triangles. Both the 4 tile and 13 tile triangles have edge shapes where two edges are the same (the “a” edge) and one is unique (“the “c” edge). Whenever two “a” edges face each other across a single row of tiles a “double diamond” ¼ tile gap occurs. Only if an “a” edge is opposite a “c” edge is there no gap. You can see this rule at work in both patterns above. The tessellating pattern can avoid all a-a facing edges but the non-tessellating pattern cannot. With a 1:1 ratio of large and small triangles it is impossible to avoid the presence of some a-a interactions so they cannot tessellate. For any proposed pattern we can thus predict how many gaps there will be by considering how many a-a contacts there are in the pattern.
The discovery of the hat shaped tile, with the recognition that it probably tessellates and never repeats is due to David Smith. The proof that he was correct is found in the comprehensive 89 page pre-print paper:
An aperiodic monotile
David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss.
Found at arxiv.doi.org/j3tw. I raise my hat to them!