Aperiodic Monotile - Hat Metatile
thingiverse
This project will help one gain an intuition on the Hat tiling algorithm described in <i>An Aperiodic Monotile</i> on page 18, figure 2.11. (https://arxiv.org/pdf/2303.10798.pdf). These Hat metatiles are designed to be printed at 0.15mm resolution with a color change at layer 1.70mm. Six 8 x 1mm magnets can be glued into place to hold the lid securely onto the base of the container. The container holds 96 H<sub>7/8</sub> metatiles (2 H<sub>377</sub> metatiles). One may consider using tape to hold metatiles together as they are being assembled. <font size="4">Background Information</font> In March of 2023 the discovery of the first aperiodic monotile (colloquially known as the “Hat”) was announced. A monotile is a single shape in which multiple copies of itself can be arranged to cover a surface without gaps or overlaps. An aperiodic monotile can tile a surface without using a repeating pattern. Tiling a set of Hats is a difficult task without knowing an algorithm for doing so. This Hat tiling algorithm begins with sets of 7 Hats (H<sub>7</sub>) and 8 Hats (H<sub>8</sub>) called Metatiles. These Hat Metatiles are assembled in a recursive fashion to produce increasingly larger coverage with each iteration. The photos show metatiles H<sub>55</sub> and H<sub>377</sub> as part of this process. This Hat tiling algorithm inflates the covered area by a factor of the Golden Ratio (φ) to the power of 4 as the metatile substitution progresses. Similarly, the ratio of right side up to upside down Hat tiles is also φ<sup>4</sup>:1. T<sub>0</sub> = 1 T<sub>1</sub> = H<sub>8</sub> = 8 T<sub>i</sub> = 7T<sub>i-1</sub> - T<sub>i-2</sub> for i ≥ 2 <table> <tr> <th>Iteration (i)</th> <th>Total Hats (T<sub>i</sub>)</th> <th>Inflation (T<sub>i</sub>/T<sub>i-1</sub>)</th> </tr> <tr> <td>0</td> <td>1</td> <td>Undefined</td> </tr> <tr> <td>1</td> <td>8</td> <td>8</td> </tr> <tr> <td>2</td> <td>55</td> <td>6.875</td> </tr> <tr> <td>3</td> <td>377</td> <td>6.85454545...</td> </tr> <tr> <td>4</td> <td>2584</td> <td>6.85411141...</td> </tr> <tr> <td>5</td> <td>17711</td> <td>6.85410217...</td> </tr> <tr> <td>Infinite</td> <td></td> <td>6.85410196... = φ<sup>4</sup></td> </tr> </table>
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