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We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging edge-to-edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles, and we apply our method to present via these rules a single prototile that can only tile ℝ3 aperiodically, and a pair of square tiles that can only tile ℝ2 aperiodically.

Aperiodic Tilings with One Prototile and Low Complexity Atlas Matching Rules