1 Construction

We attach four cubes of half the edge length diagonally to a cube, such that the contact vertices form a regular tetrahedron (Fig. 1).

Fig. 1: First Step

We then attach four more cubes to each of the attached cubes. Their edge length is one-quarter of the edge length of the initial cube. The contact vertices should again each form a regular tetrahedron, but be different from the contact vertices of the first step (Fig. 2).

Fig. 2: Second Step

Figure 3 shows further steps. A total of 7 steps are shown.

Fig. 3: The first seven steps

2 Special Views

In Figure 4, the figure with the first five steps is rotated. The outline of the figure is a regular tetrahedron.

Fig. 4: Rotation

The figure is the Sierpiński tetrahedron (a 3D version of the Sierpiński triangle), with a dual cube fitted into each recessed octahedron. The curious thing is that the cubes cannot be removed, because otherwise the entire figure disappears.