[ein Stein, 趣闻，科普] 单一形状的非周期平面密铺

英国约克郡： Yorkshire, England

滑铁卢大学： University of Waterloo

帽子： hat

乌龟： turtle

幽灵： Spectres

非周期性： aperiodic

瓷砖： tile

密铺，几何镶嵌： tiling

一、单一形状的非周期平面密铺：“幽灵 Spectres”

（1）2022-11，David Smith，英国约克郡的一位业余数学家、艺术家和密铺爱好者，退休印刷技术员（retired printing technician），找到了已知的第一种非周期性单瓷砖：“帽子 hat”。

   13边形状的帽子（13-sided shape, craggy, hatlike shape）

图1  帽子 vocabulary.png

https://www.mckennagene.com/tshirt/more/vocabulary.png

https://www.mckennagene.com/tshirt/learningMore.html

（2）几天之后，David Smith 又发现了：“乌龟 turtle”。

图2  乌龟 turtle   WithPersonality.png

https://www.mckennagene.com/tshirt/more/turtleWithPersonality.png

https://www.mckennagene.com/tshirt/learningMore.html

（3）2023-05-29 之前，David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss 找到了“幽灵 Spectres”。

   单一形状的非周期密铺完成。

图3  幽灵 spectre turtle2Spectre.png

https://www.mckennagene.com/tshirt/more/turtle2Spectre.png

https://www.mckennagene.com/tshirt/learningMore.html

（4）“幽灵 Spectres”平面密铺

图4  “幽灵 Spectres”平面密铺

Jen Christiansen A Chiral Aperiodic Monotile David Smith_裁剪_稀疏.jpg

裁剪自：

https://static.scientificamerican.com/dam/m/404b51103d7be4ff/original/saw1224Cutt39_d_TEXT.png?m=1730736763.532&w=2000

https://www.scientificamerican.com/article/mathematicians-discover-a-new-kind-of-shape-thats-all-over-nature/

（5）帽子-乌龟 动图

图5  帽子-乌龟 动图

https://cs.uwaterloo.ca/~csk/hat/examples/animation.gif

https://cs.uwaterloo.ca/~csk/hat/

二、主要发现者

图6  David Smith’s discovery has been called “mind-boggling.”

https://www.quantamagazine.org/wp-content/uploads/2023/04/Dave-Smith-courtesyofDaveSmith.webp

https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/

图7  Craig Kaplan   criag.jpg

https://images.squarespace-cdn.com/content/v1/5aa6128850a54f0331abec9a/84be0815-08d5-455a-b141-98a24bcaf02b/criag.jpg?format=2500w

https://www.mathvalues.org/masterblog/unlocking-the-aperiodic-monotiles-secrets-an-interview-with-craig-kaplan

三、相关报道

https://www.mckennagene.com/tshirt/learningMore.html

   And with the discovery of the spectre shape, the race for a true einstein was complete. This single shape is a "chiral aperiodic monotile", meaning it can tile an infinite plane, without repeating and it is truly just one shape, reflection not required.

   【机器翻译】随着幽灵形状的发现，真正的爱因斯坦的竞赛已经完成。这个单一的形状是一个“手性非周期性单调”，这意味着它可以平铺一个无限的平面，而不需要重复，它真的只是一个形状，不需要反射。

https://www.mckennagene.com/tshirt/learningMore.html

https://mp.weixin.qq.com/s?__biz=MzAwNTA5NTYxOA==&mid=2651497824&idx=2&sn=fb754f7e9262c67403332c978931d8e2&chksm=81318136ba9268c5690ba45eaea5c15e363d03f5883ca931e8b735d6b7d57e276489f30f7b23&scene=27

   2022年11月, David Smith 找到了它. 这就是“帽子”, 已知的第一种非周期性单瓷砖.

   Smith 一位业余数学家、艺术家和密铺爱好者, 他像许多数学发现那样，通过探索和观察，发现了“帽子”. 之后, Smith 与研究人员 Craig Kaplan、Chaim Goodman-Strauss 和 Joseph Samuel Myers 联系, 他们共同验证并确认了这是长久以来寻找的非周期性单瓷砖.

https://mp.weixin.qq.com/s?__biz=MzAwNTA5NTYxOA==&mid=2651497824&idx=2&sn=fb754f7e9262c67403332c978931d8e2&chksm=81318136ba9268c5690ba45eaea5c15e363d03f5883ca931e8b735d6b7d57e276489f30f7b23&scene=27

https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile

   Although the name “einstein” conjures up the iconic physicist, it comes from the German ein Stein, meaning “one stone,” referring to the single tile. The einstein sits in a weird purgatory between order and disorder. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can’t form a pattern that repeats.

   【机器翻译】虽然“爱因斯坦”这个名字让人联想到这位标志性的物理学家，但它来自德语ein Stein，意思是“一块石头”，指的是一块瓷砖。爱因斯坦坐在秩序和混乱之间的一个奇怪的炼狱里。尽管瓷砖整齐地组合在一起，可以覆盖无限的平面，但它们是非周期性的，这意味着它们不能形成重复的图案。

https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile

参考资料：

[1] 中国科学院物理所，2024-09-27，探索密铺的奥秘：从平移对称到非周期单元

https://mp.weixin.qq.com/s?__biz=MzAwNTA5NTYxOA==&mid=2651497824&idx=2&sn=fb754f7e9262c67403332c978931d8e2&chksm=81318136ba9268c5690ba45eaea5c15e363d03f5883ca931e8b735d6b7d57e276489f30f7b23&scene=27

[2] David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss. A chiral aperiodic monotile [J].  Combinatorial Theory, 2024, 4(2): 

https://escholarship.org/uc/item/4xn41982

https://arxiv.org/abs/2305.17743

[3] Science News, 2023-03-24, Mathematicians have finally discovered an elusive ‘einstein’ tile

https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile 

[4] Erica Klarreich, 2023-04-04, Hobbyist Finds Math’s Elusive ‘Einstein’ Tile

https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/

[5] The University of Waterloo, 2023-07-05, The vampire einstein: Researchers discover a single shape that tiles the plane aperiodically without reflection

https://cs.uwaterloo.ca/news/vampire-einstein-aperiodic-monotile-without-reflection

动图 hat-tiling-animation.gif

https://cs.uwaterloo.ca/sites/default/files/uploads/images/hat-tiling-animation.gif

https://cs.uwaterloo.ca/~csk/hat/examples/animation.gif

[6] 知乎，2023-10-23，花费数学家400多年时间的非周期密铺问题

https://www.zhihu.com/zvideo/1699801100004417536?utm_psn=1845834964324712448

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