Abstract:In this paper, we prove that it is undecidable whether a set of two polycubes can tile $\mathbb{Z}^3$ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By expanding this technique to higher dimensions, we also prove that a set of disconnected tiles in $\mathbb{Z}^n$ can be simulated by the same number of connected tiles in $\mathbb{Z}^n$ for $n \geq 3$.
From: Yoonhu Kim [view email]
[v1]
Fri, 15 Aug 2025 06:33:49 UTC (21,961 KB)
[v2]
Mon, 10 Aug 2026 14:04:16 UTC (4,258 KB)
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