Connected Holes

Abstract

It is possible to make interwoven structures by using two or more surfaces with holes. Several solutions are known. And also interwoven structures have been made with two or more 3D objects with holes.One example is M.C. Escher’s ‘Double Planetoid’ [1], a combination of two tetrahedra. In this paper I show that we don’t need two or more separate surfaces: even with one single surface we can make interesting interwoven structures which seem to be multi-layered, in 2D as well as in 3D.

1. Introduction

1.1. Interwoven Surfaces. In Figure 1 an example of a structure consisting of two interwoven surfaces is shown. In both layers holes are made at such places that the surfaces can be woven. In all three interwoven layer structures of Figures 1, 2 and 3 the midpoints of the holes are placed on a square grid. The shapes of the holes in the layers are respectively rounded, elliptical and hexagonal. In addition to the shape of the holes, the weaving in the three examples is also different. The design of the interwoven structure in Figure 3 is made by M.C. Escher [2].
Figure 1: Interwoven surfaces - a
Figure 2: Interwoven surfaces - b
Figure 3: Interwoven surfaces - c
1.2. Hexagonal Grid. We can use almost any pattern as an underlying grid for an interwoven layer structure. A hexagonal grid is used in the examples of Figure 4 and Figure 5. Also the number of layers may vary. A structure of three interwoven layers is shown in Figure 5.
Figure 4: 2 Interwoven surfaces
Figure 5: 3 Interwoven surfaces
1.3. 3D Structures. Besides the interwoven 2D layer structures there are also structures with interwoven 3D objects. As an example we can take M.C. Escher’s woodcut ‘Double Planetoid’. “Two tetrahedra going through each other, gliding through space as a planetoid. Together both objects are one connected structure, but they do not know of each others’ existence”, Escher said about this print [1]. And we can also say this about the two Möbius bands in Figure 6, or the two spheres in Figure 7.
Figure 6: Entwined Möbius bands
Figure 7: Entwined spheres