Aperiodic Monotiles - Hat and Spectre

thingiverse

These three shapes are known in the mathematics community as “Hat” and “Spectre” tiles. The challenge is to arrange these sets of tiles without any gaps, or overlaps. Hat (red)   No repeating pattern with upside down tiles allowed Straight Sided Spectre (green)   No repeating pattern with right side up tiles only   No repeating pattern with upside down tiles allowed   Repeating pattern with upside down tiles allowed Spectre (blue)   No repeating pattern with right side up tiles only Each container holds roughly 100 tiles. You may want to consider changing colors halfway through printing the tiles so that their orientation can be easily identified. Six 8 x 1mm magnets can be glued into place to hold the lid securely onto the base of the container. <font size="4">What is an Aperiodic Monotile?</font> A set of geometric shapes which completely cover a surface without gaps or overlaps is known as a tiling. If a tiling can be translated and/or rotated to be an exact match with the original, then the tiling is called periodic (repeating). So an aperiodic monotile is a single shape that can be tiled to completely cover a surface without repeating. Before March 2023 it was unknown whether an aperiodic monotile existed. Then “The Hat” was discovered. The Hat tiling requires 1 tile to be flipped upside down (reflected) for roughly every 7 right side up tiles. The Golden Ratio to the power of 4 is the precise ratio of unreflected vs reflected tiles. Folks quickly set out to eliminate the need for reflecting tiles. Within about a month “The Spectre” was derived from the Hat. The Spectre is also aperiodic, but does not require reflections to completely tile a surface. The Hat and Spectres are closely related. The sides of the Hat are of lengths 1 and sqrt(3). If all sides of the Hat are made to be of length 1 while preserving the interior angles, a straight sided Spectre tile is produced. Consider these three characteristics of tilings: <ol> <li>Aperiodic</li> <li>Reflections</li> <li>Periodic</li> </ol> The Hat has characteristics 1 and 2, while the straight sided Spectre has 1, 2, and 3. By changing each side of the straight sided Spectre into a curve, characteristics 2 and 3 disappear leaving an aperiodic monotile with no reflections. These are the original papers publishing this significant discovery. Hat: https://arxiv.org/pdf/2303.10798.pdf Spectre: https://arxiv.org/pdf/2305.17743.pdf