On 2-reptiles in the plane
The authors study \(n\)-reptiles in \(\mathbb{R}^2\), that is planar bodies \(T\) such that \(\sqrt{n}T=\bigcup (e^{i\theta_j}T+a_j)\) \((j=1,\dots , n)\). An \(n\)-reptile is rational, if the numbers \(\theta_j-\theta_k\) all are rational. Theorem 1.1: Let \(T\) be a rational \(n\)-reptile. Then each number \(\sqrt{n} e^{-i\theta_j}\) is either an integer or a non-real quadratic integer. In Theorems 1.2, 1.3 all 6 (equivalence classes of) rational \(2\)-reptiles are determined. Theorem 1.4: There is no rational \(2\)-reptile which is also a rational \(3\)-reptile. Conjecture: All reptiles are rational.
scientific article; zbMATH DE number 1547917