7. Infinite Polyhedra

7.1. Cubic ring structure. There is another way of creating three dimensional entwined circular ring structures. And in some sense this way is more connected to Escher's search. We start with a set of rings in a cubical space frame structure as in Figure 28 in which you can recognize Escher's first drawing of display A. Now we add a second set of rings to connect all the rings. This second set has the same structure as the first one but can also be seen as the dual set.
Figure 28: Cubical ring structure.

7.2. Tetrahedral ring structure. Starting with the first entwined ring pattern of display B (Figure 11) we get the tetrahedral entwined ring structure of Figure 29.
Figure 29: Tetrahedral.

7.3. Triangles and squares. In the structure of Figure 29 we can distinguish subsets of 4 rings each. The rings of such a subset are lying on the triangular faces of a tetrahedron. It might be a surprise that, seen from a certain angle, the structure looks exactly the same as the pattern of Figure 21, which is based on a square grid (Figure 31). And something similar occurs when we take the cubic ring structure of Figure 28: seen under the right angle we will see threefold symmetry.
Figure 30: Tetrahedral - threefold symmetrie
Figure 31: Tetrahedral - twofold symmetrie

8. Conclusion

8.1. Escher's research. Somehow it's a pity that we do not know how Escher would have continued his research. But the first display of entwined circular rings already appears to be very inspiring and opens up many directions for further research.

References

[1] Doris Schattschneider, M.C. Escher: Visions of Symmetry, Abrams 2004
[2] Herman J. de Vries, Meetkunstig Vlakornament, De gebroeders van Cleef 1891
[3] M.C. Escher, Notebooks (unpublished), Haags Gemeentemuseum