6. Borromean Polyhedra

6.1. Mirrored Corners. The Borromean Joint can be used to create polyhedral constructions. The first example is the cube. The cube in Figure 45a is made of twelve straight elements with holes at both ends. When you look close you will see that two neighbour connections are each others’ mirror image. When we want to have each of the connections exactly the same we have to use either curved elements (Figure 45b) or bent elements (Figure 45c).
Figure 45a: Borromean Cube.
Figure 45b: Curved elements.
Figure 45c: Twisted elements.
Figure 46: Tetrahedron – curve elements.
Figure 47: Tetrahedron – twisted elements.
6.2. Equal Corners. The next example is the tetrahedron. Because of the odd number of edges of each of the faces we can not have two different corner connections as in the cube of Figure 45a. So we have to either use the curved elements (Figure 46) or the twisted elements (Figure 47). This is also the case when we want to construct the dodecahedron (Figure 48b). Each face of the dodecahedron has five edges which is an odd number. The biggest number of elements that can be slid together without problems automatically creates an Hamilton path on the dodecahedron (Figure 48a). In Figure 49 you can recognise the truncated icosahedron. Also here it is necessary to use bent elements.
Figure 48a: Hamilton path.
Figure 48b: Dodecahedron.
Figure 49: Tr. Icosahedron.
6.3. Straight Elements. Besides the cube there are three more polyhedra which only have faces with an even number of edges: the truncated octahedron (Figure 50), the rhombitruncated cuboctahedron (Figure 51) and the rhombitruncated icosidodecahedron (Figure 52).
Figure 50: Polyhedron 1.
Figure 51: Polyhedron 2.
Figure 52: Polyhedron 3.
6.4. Borromean Rings. In Section 5.3 we have seen that we can make more than one Borromean Ring connections with each ring in a standard Borromean Ring pattern. In the examples below you can see how this idea can be used to build polyhedral with just rings. The tetrahedron (Figure 53) and the dodecahedron (Figure 55) are now build using rapid prototyping techniques and in the models the rings are not connected but it is not possible to change the structure.
Figure 53a: Single Ring.
Figure 53b: Three Rings.
Figure 53c: Tetrahedron.
Figure 54: Borromean Cube.
Figure 55: Borromean Dodecahedron.
7. Conclusion

7.1. Conclusion. I think we may say that ‘Through and around weaving’ is as inspiring as ‘Over and under weaving’. Escher did some experiments in this field most probably inspired by the Italian iron window lattices. Nice constructions can be made with this technique especially in combination with the Borromean ring structure.

References

[1] Rinus Roelofs, Het onmogelijke tralieraam, in Pythagoras, 1998.
[2] D. Schattschneider, M. Emmer (Eds.), M.C. Escher’s Legacy, Springer Verlag, 2002.
[3] J.L. Locher, W.F. Veldhuysen, The Magic of M.C. Escher, Thames & Hudson, 2000.