An aperiodic pair of tiles in \(\mathbb{E}^n\) for all \(n\geq 3\) (Q1304433)

A set of tiles in \({\mathbb E}^n\) is called aperiodic if any tiling of the space with copies of these tiles is non-periodic. The author gives a set of two tiles, which is proved to be aperiodic. One of the tiles has a disconnected interior. When a connected interior is required, the following aperiodic sets are given: a set of two tiles in \({\mathbb E}^3\), a set of three tiles in \({\mathbb E}^n\), \(n > 3\), and a set of two tiles in \({\mathbb E}^n\), \(n > 3\), with a special matching rule. The author has published a separate paper devoted to the case \(n = 2\) [ibid., 375-384 (1999; see the paper above)].