Fuglede's conjecture is false in 12 and higher dimensions

We give an example of a set $\Omega \subset \R^{12}$ which is a finite union of unit cubes, such that $L^2(\Omega)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\Omega|^{1/2}} e^{2\pi i \xi_j \cdot x}: \xi_j \in \Lambda \}$ for some discrete set $\Lambda \subset \R^{12}$, but which does not tile $\R^{12}$ by translations. This answers a conjecture of Fuglede in the negative, at least in 12 and higher dimensions.