But we can continue the process and extend the faces a little further (Figure 10) The second position where we have a polyhedron again is shown in Figure 11.
Figure 10: Development of the shape of the faces for the second stellation of the icosahedron.
Figure 11: Second stellation of the icosahedron.
Just like in Escher’s print “Gravity” I have removed parts of the faces to get a good idea of the construction. I was fascinated by the interesting structure that appeared now on top of all faces. Not one pyramid but an intersection of 3 somewhat deformed pyramids. So one step more in the stellation process leads to very interesting structures. The question now arose if something similar would be possible with Elevation. Can we define second elevation? And what kind of objects can we expect then?

2.3. Second Elevation. I used the generalized elevation process defined in section 2.1 and applied this to the octahedron, elevating the midpoint of each face of the octahedron until the new shape was similar to another polyhedron. The first polyhedron gotten this way was the rhombic dodecahedron. On this polyhedron we can apply the same process again and call the next result a second elevation of the octahedron. We stopped at the point that we got coplanar sets of elevated faces on the faces of the rhombic dodecahedron (Figure 12).
Figure 12: First and second elevation of the octahedron.
The end figure (Figure 13a) now appears to be the same as one of the polyhedra Escher constructed (shown in his print " Studie voor de houtgravure Sterren (1948)) and later (1961) used in his print “Waterfall” as an ornament at the top of one of the towers (the closed version as in Figure 13b).
Figure 13a: Escher's polyhedron.
Figure 13a: Escher's polyhedron in "Waterfall".