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[ein Stein, 趣闻，科普] 单一形状的非周期平面密铺
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[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
图4 Figure 1.1: The 14-sided polygon Tile(1, 1), shown on the left, is a weakly chiral aperiodic monotile: if by fiat we forbid tilings that mix unreflected and reflected tiles, then it admits only non-periodic tilings. By modifying its edges, as shown in the centre and right for example, we obtain strictly chiral aperiodic monotiles called “Spectres” that admit only non-periodic tilings even when reflections are permitted.
https://escholarship.org/uc/item/4xn41982
（4）“幽灵 Spectres”平面密铺
图5 “幽灵 Spectres”平面密铺
Jen Christiansen A Chiral Aperiodic Monotile David Smith_裁剪_稀疏.jpg
裁剪自：
（5）帽子-乌龟 动图
图6 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7 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8 Craig Kaplan criag.jpg
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
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://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
[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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