2. Elevation versus Stellation

2.1. Difference. There is a fundamental difference between the elevation of the octahedron and the stellation of the octahedron: the total number of faces of the elevation is 32, whereas the number of (intersecting) faces of the stellation is 8. And yet there is another important difference. Kepler, in his definition, talks about a process. When we read the descriptions of Pacioli, he is just talking about the final result. We can however redefine “Elevation”, introducing the process as follows: Elevation for polyhedra is the process of pulling each midpoint of all of the faces outwards until the triangles formed by those midpoints with two adjacent vertices of the original face form are equilateral). A generalization can be made by not to demand that the triangles must be equilateral. With this definition Pacioli’s elevated dodecahedron can be seen as a step in between the dodecahedron and Kepler’s first stellation of the dodecahedron. In the print “Gravity” by M.C. Escher (Figure 5) parts of the star shaped-faces are removed, which gives us a good look at the construction. Escher writes about this print: “On each of the twelve faces we can see a monster of which the body is captured under a five sided pyramid.”. Which comes close to Pacioli’s description of the elevated polyhedra. Using the new definition of elevation, we can compare the process of development from dodecahedron to the elevated version with the development from dodecahedron to Kepler’s stellation. In the pictures (Figure 6 and 7) only 6 faces (the upper part) of the dodecahedron are shown. In the elevation process, the third step shows the object that is published in “La Divina Proportione”, but when you continue the process of elevation you can end up at the state that is similar to the stellated dodecahedron, since in this state, five triangular faces of the elevated figure are coplanar with an original face of the dodecahedron.

Figure 6: Elevation.
Figure 7: Stellation.
The big difference between the two end objects is the number of faces counted. In the elevated version we still have the pentagonal faces inside, so both objects are double layered.
2.2. Second Stellation. Starting with the icosahedron we can extend the triangular faces to create a stellated icosahedron. Four steps of this dynamic process are shown in Figure 8. In the process of stellation this is the first position in which we get a new polyhedron (Figure 9).
Figure 8: Development of the shape of the faces for the first stellation of the icosahedron.
Figure 9: First stellation