3. Edge Elevation

3.1. Edge Elevation. Studying the end figure a little more brought me to the conclusion that you also can see this as an octahedron with a somewhat deformed rhombic pyramid on each of the edges.
Twelve pyramids surround the octahedron. That means that we can reach the same end figure by elevating the midpoint of each of the edges. So we can define a new transformation: “Edge Elevation”, in which the midpoint of each edge is connected to the midpoint of the faces that meet that edge and then the edge midpoint is pulled outward, stretching its connection to form a pyramid of four triangles.
Figure 14: Edge Elevation of the Octahedron
Figure 15: Edge Elevation of the Octahedron - open
Figure 14 illustrates the process. Figure 15 shows the same process but now uses Escher’s way of opening up the structure as in his print “Gravity” (Figure 5).

3.2. Tetrahedron and Icosahedron. When we apply edge elevation to other polyhedra like the tetrahedron and icosahedron we will see the developments of the objects as in Figure 16 and 17. The final state of the development of the edge elevated icosahedron is similar to the second stellation of the icosahedron. In the case of the tetrahedron we end up with a new object.
Figure 16: Edge Elevation of the Tetrahedron - open
Figure 17: Edge Elevation of the Icosahedron - open
4. Construction

4.1. Construction Elements. The final state of the edge-elevated icosahedron looks similar to the second stellation of the icosahedron but there is big difference. When we analyze the construction of both objects we will see that their faces are different. At left in Figure 18 is one of the faces of the second stellation of the icosahedron, at the right two connected faces of the edge-elevated icosahedron. To build models of the edge-elevated polyhedra I laser-cut the faces in pairs with a folding line on the common edge. The paper models appeared relatively simple to build.
Figure 18: Faces of the second stellation of the icosahedron and of the edge-elevated icosahedron.