Schmitt–Conway–Danzer biprism

The , or SCD prototile, is a convex polyhedron which can tessellate 3-dimensional space but only aperiodically. It is abstractly equivalent to the gyrobifastigium.

Solution to the einstein problem
The can be made to tessellate 3-dimensional space and no tiling of just s has any translational symmetry. However, some of these tilings have screw symmetry, that is symmetries which are a simultaneous rotation and translation. These tilings are thus aperiodic but not strongly aperiodic. This makes the a solution to some versions of the einstein problem, which do not require the strong version of aperiodicity.

The along with its mirror image can tile the plane periodically.

History
A non-convex variant was originally discovered by Schmitt in 1998. Schmitt conjectured that it could be made convex while maintaining its properties as a prototile. Conway found such a variant and in 1993 Danzer improved the design further.