Some unusual space-filling solids

Extract

When identical copies of a solid can be packed together indefinitely to cover the whole of space and leave no gaps it is said to be space-filling. Usually only convex solids are considered in this context since, as Grünbaum and Shephard point out, non-convex space-filling solids can be generated from convex examples by making compensating additions and reductions, analogous to Escher’s modifications of plane tessellations. Nevertheless some non-convex space-filling solids are not obvious derivatives, and are interesting in their own right. Although the examples considered here have all been generated in the same way (to be described), their relation to other space-filling objects will also be considered.