4. "Tile Rotation"

4.1. Bending the pieces. There is one special group 2D slide-together structures that can only be assembled when we allow bending the pieces during the assembly. I decided to add this group because of a very interesting property. The pieces of this group consist of two connected triangles with both two halfway cuts (Figure 23) and in the final structure the double connection is used (Figure 24). Because now the pair of slide lines is not a parallel pair, bending of the pieces is needed.
When we look at the final structure we can recognize a tiling pattern lying on the surface: a pattern with small and big square tiles (Figure 25). And when we look through the structure we can recognize the same tiling but now with the opposite orientation: left turning instead of right turning.

When we project the upper tiling onto the lower tiling we see that the center of each tile is in the same position and each tile seems to have been rotated around its own center. I found a few other tilings to which I could apply this operation, which I called 'tile rotation'. I wanted to examine whether it was possible to construct a slide-together structure from tilings with this property.
In the tiling of Figure 26 we can change the orientation by rotating the hexagons and the triangles, each around their own center. And so from this tiling I was able to design the accompanying slide-together structure (Figure 27 and 28).

5. Mortise-and-tenon Joints

5.1. Introduction. Another way of making slide-together structures is the use of mortise-and-tenon joints. The structure in Figure 29 consists of six equal elements and has some similarities with the ring structure of Figure 2. Here too we have got two possibilities. You either take it apart into two groups first or you take it apart by moving all the elements away from the center at the same time. And to become an infinite structure we can again double the elements. See Figure 30.