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2 Squares.
These tiles produce the same tiling as C. Goodman-Strauss's
Trilobite and Cross. But the first set depicted denies the
fault-line that the other and Goodman-Strauss's permits. The tile
with corner tags may be omitted and the other tiles made square to
compensate (keeping the bumps and notches) while the stripes are
interpreted as crossing vertices and having no lateral dimension.
I'll leave the reader to become aquainted with Goodman- Strauss's
proof in A small aperiodic set of planar tiles, Europ. J. Combinatorics, 20 (1999) 375-384. A draft is available on his website.
For interest I show a fault-line in 2 Squares fault 1 and 2 Squares fault 2.
A red stripe is represented by a dashed line and a blue stripe by
a solid line. Further mismatches are simultaneously possible at a
greater distance on either side of the centre of the fault-line.
A tiling without a fault-line is depicted in 2 Squares tiling. |