5. M.C. Escher Stars

5.1. M.C. Escher's 'Gravity'. In the print ‘Gravity’ M.C. Escher has used a star-shaped object on which twelve animals are situated [1]. For each animal there is a floor to stand on and a roof to cover. The floor and the roof can be seen as two layers. But the complete object is not a combination of two objects as in M.C. Escher’s print ‘Double Planetoid’ [1]. In this object (Figure 27) the star-shaped planes are connected in such a way that we can walk from any plane to any other plane.
5.2. Interwoven Layer in 3D. The star-shaped object M.C. Escher has used in his print is one continuous surface, but we can see clearly that the object has two layers. This is in fact the same situation as we have seen in the interwoven layer structures in section 3. To make this clearer, we can enlarge the holes of M.C. Escher’s object as in Figure 28. And when we zoom in to one of the holes (Figure 29) we recognize the same trefoil knot we have used in the objects shown in paragraph 4.1. In fact after rounding the tops of the pyramids of M.C. Escher’s object we will get the same object as shown in Figure 25.
Figure 27: Escher's object
Figure 28: Enlarged holes
Figure 29: Trefoil knot
5.2. Stellation. Star-shaped objects can be created starting from a Platonic or an Archimedean Solid. As an example, this process is shown in Figure 30 where we start with the icosahedron (Figure 30a). In the first three steps (Figures 30b, 30c, and 30d) we extend the triangular faces of the icosahedron until they touch each other again. In Figure 30d this process is completed and the result is a star polyhedron. Note that this object has two layers: beneath each pyramid there is still the original triangular face. We will make this visible by cutting holes in the pyramids (Figures 30e and 30f). And in the final steps (Figures 30g, 30h, and 30i) we enlarge the holes so that the shape of the hole, which is in this case a fivefold knot, can be seen clearly. The result is an interwoven structure built with one continuous surface (Figure 31).
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Figure 30: Stellation of the icosahedron
Figure 31: Double layer structure based on the icosahedron