Schlegel Diagrams

Abstract

We have, throughout the previous chapters, repeated ad nauseam two principles: (1) Polyhedral surfaces are two-dimensional, and (2) we are not committed to definite distances and angles, but only to numbers of elements, their dimensionalities, and their valencies. Given these two principles one concludes that every polyhedron may be distorted such that it can be laid out flat on a surface so that no edges cross. This is done by choosing one particular face, extending it such that it becomes the frame within which the remainder of faces, edges, and vertices are contained. Visually this amounts to holding one face quite close to the eyes, looking at the structure through that face, and drawing the projection of the structure as seen in this exaggerated perspective. (Note that perspective actually means “as seen through”!) Such a perspective projection of a polyhedron is called a Schlegel diagram. For our discussion Schlegel diagrams are important because they do not just represent our structures: they are our structures. Although we find it convenient to compare a structure having eight 3-valent vertices, six 4-valent faces, and twelve edges to the cube, so familiar to us, it can be equally well represented by the Schlegel diagram of Fig. 8-1, which has exactly the same elements and valencies as does the cube. The advantage of the Schlegel diagram is that all its elements and connections are explicitly visible, with no hidden elements. The fact that the face represented by the circumference of the Schlegel diagram must be counted explicitly as a face certainly need not be belabored in the context of this treatise (cf. Chapter 5).