2. Knots

2.1. Elevation of the Cube. After analyzing the double layer structure of Leonardo’s elevated octahedron, the question might be whether we will see something similar at the other elevated polyhedra. In Figure 10 Leonardo’s drawing of the elevated cube is shown. Luca Pacioli explains: “… it is enclosed by 24 triangular faces. This polyhedron is built out of 6 four-sided pyramids, together building the outside of the object as you can see it with your eyes. And there is also a cube inside, on which the pyramids are placed. But this cube can only be seen by imagination, because it is covered by the pyramids. The 6 square faces are the bottom faces of the 6 pyramids.”. When we follow this explanation we can make the exploded view presented in Figure 11.

Figure 10: Elevated Cube.
Figure 11: Elevated Cube.
Figure 12: Elevated Cube.
Figure 13: First Loop.
Figure 14: Two separated loops.
Looking at the vertex figure around the one of the vertices of the original cube we see again two separated loops, when we follow the description of Luca Pacioli. And these loops can be described as 333333 plus 444. The steps are shown in Figure 12, 13 and 14. Also in this case we have the situation that four faces come together at the edges that meet in these vertices. So also here we can choose an alternative order. Instead of just following the outside shell or the inside shell we start at one of the faces of the cube and then step over to the adjacent triangle face and after a second triangular face we step back on a face of the cube again. The total order of the faces of our walk will be 433433433 (Figure 15).
Figure 15: Alternative path.
Figure 16: Trefoil knot.
Figure 17: Single surface.
But that means that we now have one single loop that makes two tours around the vertex. We can modify the shape of the loop so that, when we use it as a cutting line in the total construction, all edges are connecting lines between just two faces (Figure 16). The total shape of the loops then will show the trefoil knot (Figure 17). By definition the “elevation element” for the elevated cube is the object shown in Figure 18. For the real physical construction we cut away some parts following the knot-line. And now the final model can be build with 6 of these elements. The end result is a single surface forming a double covering around the cube.
Figure 18: Elevation element.
Figure 19: Construction element.
Figure 20: ”Double Cube”.