Disk-like Tiles Derived from Complex Bases

For each positive integer k, the radix representation of the complex numbers in the base-κ + i gives rise to a lattice self-atone tile Tκ in the plane, which consists of all the complex numbersthat can be expressed in the form ∑j>1 dj(-κ + i)^-j, where dj ∈ {0, 1,2 , κ^2}. We prove that Tκ is homeomorphic to the closed unit disk {z ∈ C : |z| ≤ 1} if and only if k ≠ 2.