# Aperiodic Polycube Tiling(s).

In a sense the 3D chair tiles may be viewed as hepta-cube tiles. Here we we will demonstrate similar aperiodic substitution tiling based on two types of poly-cubes, the tri-cube and tetra-cube shown in figure 1.

For a multiplication factor of 2, the first substitution tiles contain 24 and 32 small cubes respectively. The bottom and top face of the tricube are identical to the 2D chair tile. Therefore, a trivial tri-cube substitution tile consists of two layers of four tri-cube prototiles. Also a trivial tetra-cube supertile can be built from 8 tetra-cube prototiles by combining two of them to a basic cube. Therefore, we demand that the substitution tiles are built from a finite number of both types of prototiles, infact we choose the numbers are four and three respectively for the super-tri-cube, and four and five for the super-tetra-cube. There are still a number of different ways to construct the supertiles. Our choise is to make them as symmetrical as possible as shown in figure 1.