A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

翻译：一个长期未解的公开问题要求找到一种非周期单瓷砖，也称为“爱因斯坦瓷砖”：一种能够铺满平面但永远不会周期性铺满的形状。我们通过展示一个连续族组合等价的非周期多边形，解答了拓扑圆盘瓷砖的这个问题。我们首先证明一个代表性例子——“帽子”多风筝形——可以形成称为“元瓷砖”的簇，并为这些簇定义替换规则。由于元瓷砖可以铺满平面，帽子形同样可以。接着，我们通过一种新的几何不可公度性论证，证明该连续族中的一般多边形是非周期的。此外，我们给出一个组合的、计算机辅助的证明，表明帽子形必须形成分层的——因此是非周期的——铺砌。