2024/12/14 by Chan Yang, Yang, Chan, Zhujun Zhang +1 · 2 citations
Computer Science · #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2412.10646
Is there a fixed dimension n such that translational tiling of ℤn with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension n [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension n is part of the input, then the translational tiling for subsets of ℤn with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of ℤn with a monotile is undecidable, for some fixed n. This paper gives another supportive result for this conjecture by showing that translational tiling of the 4-dimensional space with a set of three connected tiles is undecidable.