Aperiodic Monotile - 帽形元瓷砖

Aperiodic Monotile - Hat Metatile

2023-08-02 07:23:25

使用说明

所有模型资源均由用户上传分享，内容来源于网络公开资源
侵权投诉：通过抖音私信 @jobsfan 联系我们（需附版权证明），24小时内处理
模型将通过邮件发送（5分钟内自动发送），感谢理解带宽压力，超过1小时没收到，请联系我们

关于费用

我们是爱好者共建社区，为维持服务器成本，每个模型收取微量费用（仅覆盖基础开支）。我们承诺最低成本运营，感谢您的支持！

扫码手机访问

抖音私信 @jobsfan

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