Wild tiles in $\mathbb R^3$ with spherical boundaries

Wildly embedded tiles in $\mathbb R^3$ with spherical boundary are discussed. The construction of the topologically complicated, crumpled cube tiles is reviewed. We construct an infinite family of wildly embedded, cellular tiles with Fox–Artin-type wild points. Finally, a condition on the set of wild points on a cellular tile is given to show that certain wild cells cannot be tiles. Several observations are recorded for further investigations.