7. Helical Holes

7.1. Twist again. So far we have seen that ‘twist’ is the base of some powerful operations to generate new weaving patterns as well as new weaving structures. Until here all the operations were applied on line or loop weaves. But we can also define a twist operation that can be applied on weave surfaces. In Figure 37a you can see a part of the weave surface that is created in Figure 28. It is a part that surrounds one hole in the surface. Or the part that is connected to one loop in the original weave. When we split up this part into four pieces (this is the number of surrounding holes) we can twist one part over 360/4 degrees. In Figure 37b you can see that the edge of the hole is now transformed into a non-closed line and after twisting one more part this line is transformed into a helix (Figure 37c). The unit we now have created can be used to create a surface as shown in Figure 37d. In fact such a surface can be extended into infinity in all directions. And special about this surface is that it has helical holes.
Figure 37a: One hole
Figure 37b: First twist
Figure 37c: Helix
Figure 37d: Surface
7.2. Weaving with Helical Hole Surfaces. Another property of these surfaces is that they can be used to make weaves again. We can weave two or more of these surfaces through one another. And there are also several ways to do this. In Figure 38b an extra surface is added which is moved horizontally to create a weave that can be compared with the weave in Figure 28d. The other way of weaving is adding a copy which is translated in the vertical direction (Figure 39). Both operations can be combined to create a four-layer weave with helical hole surfaces. As you can see in Figure 39d there is still space to add more layers in between the two layers. The number of possible layers depends on the thickness of the surface and the pitch of the helix.
Figure 38a: Surface
Figure 38b: Weaving a
Figure 39a: Surface
Figure 39b: Weaving b
7.3. Examples. Any tiling that can be coloured as a checkerboard can be transformed into a weaving pattern, but only when the degree of all the vertices is four. It turns out that all such tilings also can be transformed into spiral hole surfaces, but now also for degrees of the vertices higher then four. In Figure 40 we see a helical hole surface based on the Archimedean tiling (3.3.3.3.3.3), a tiling with vertex degree 6. Figure 40c and 40d respectively show the weaving of two and three surfaces. The helical hole surface of Figure 41 is based on the Archimedean tiling (3.4.3.6).
Figure 40a: Surface
Figure 40b: One layer
Figure 40c: Two layers
Figure 40d: Three Layers
Figure 41a: Surface
Figure 41b: One layer
Figure 41c: Two layers
Figure 41d: Three Layers