The Concept of Elevation applied to Flat Tiling Patterns

Abstract

In their book “La Divina Proportione”, Luca Pacioli and Leonardo da Vinci described and illustrated an operation which you can apply to a polyhedron, called Elevation. In this paper I want to show how you can make models of these elevations from simple elements, material that can be used for workshops. Luca Pacioli and Leonardo da Vinci applied the elevation operation only to polyhedra, but the concept can also be applied at 2D tiling patterns, resulting in interesting single and double weaving patterns.

1. Introduction

1.1. Elevation. In La Divina Proportione [1] by Luca Pacioli and Leonardo da Vinci, an interesting new concept that can be applied on polyhedra is introduced. The concept of Elevation that they introduced was also the subject of my paper for Bridges in 2014 [2], in which the difference between Elevation and Stellation was explained.
Figure 1: Elevated Cube
Pacioli doesn’t give a real definition of “Elevation” but his descriptions are very clear: he describes the elevated version of the cube (Figure 1) as follows: “… 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.” [3]. There is another way to describe the process: 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 demanding that the triangles must be equilateral.
1.2. Building Models. The way Leonardo da Vinci made the drawings of the elevated polyhedra suggest that there were real hanging models. And indeed it is very inspiring to have real models of these constructions. Therefore I started to try to find an easy way to make paper models of the designs presented by Luca Pacioli and Leonardo da Vinci. The Elevation of a polyhedra is not just a collection of pyramids, as in Figure 2), but there is also something inside. It is in fact a double layer construction. I had to find a way to keep this visible. The solution for this was found in M.C. Escher’s print “Gravity”, a drawing of a stellated dodecahedron (Figure 3). So by opening up the pyramids I succeeded in developing simple basic elements (Figure 4) that could be made of paper, and still allowing a look at inside layer of the construction.
Figure 2: Elevation of triangle, square and pentagon.
Figure 3: Escher's star.
Figure 4: Developing the elements for the models of the elevated polyhedra.