We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar–Taylor monotile, but can be realised by shape alone. The second is a dendrite rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our dendrite rule initiates a new method to produce tilings of the plane.

中文翻译：

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非周期性单晶通过树突强制非周期性

我们介绍了一种新型的非周期性六角形单面砖；允许无限多个平铺的原型，但任何此类平铺都缺乏任何平移对称性。将我们的Monotile副本添加到图块补丁中，必须满足两个仅适用于相邻图块的规则。第一个灵感来自Socolar–Taylor单片砖，但仅靠形状即可实现。第二个是树突规则；只要单块砖上的树与相邻砖块之一上的树连续连接，我们的单块砖的直接等距图可以添加到任何砖块中。这种情况迫使瓷砖沿着树枝状晶体生长，最终导致非周期性瓷砖。我们的枝晶规则启动了一种新方法来生成平面的平铺。