Aperiodic Monotile - 帽形元瓷砖

Aperiodic Monotile - Hat Metatile

2023-08-02 07:23:25

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Summary

This project will help one gain an intuition on the Hat tiling algorithm described in An Aperiodic Monotile 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_7/8 metatiles (2 H₃₇₇ metatiles). One may consider using tape to hold metatiles together as they are being assembled.

Background Information

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 (H₇) and 8 Hats (H₈) called Metatiles. These Hat Metatiles are assembled in a recursive fashion to produce increasingly larger coverage with each iteration. The photos show metatiles H₅₅ and H₃₇₇ 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 φ⁴:1.

T₀ = 1
T₁ = H₈ = 8
T_i = 7T_i-1 - T_i-2 for i ≥ 2