3 ms·
The spectre has 13 sides. Is it known whether this is the minimum possible for an einstein that only tiles aperiodically and without reflection?
by tromp 3y ago
The spectre has 13 sides. Is it known whether this is the minimum possible for an einstein that only tiles aperiodically and without reflection?
- robinhouston 3y agoI'm sure it is not known. We know almost nothing about this class of tiles, except for the very recent discovery that the class is non-empty.
- bazzargh 3y agoThe original monotile paper https://arxiv.org/abs/2303.10798 https://arxiv.org/abs/2303.10798 mentions that (at the time that was written) the minimum number of sides had to be 5 <= n <= 13, but also that a smaller tile can't be a polykite like the hat; exhaustive search had shown that there were no other aperiodic monotiles (than the hat and turtle) with < 21 kites But a correction: the basic shape of the spectre is 14 sides with 2 of them colinear. It's only truly aperiodic when you modify the 14 edges, making it clear it is 'really' 14 sided not 13.