# Are Your Polyhedra the Same as My Polyhedra?

• Branko Grünbaum
Chapter
Part of the Algorithms and Combinatorics book series (AC, volume 25)

## Abstract

“Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space-as we shall do from now on -“polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht Dürer [[17]] in 1525, or, in a more modern version, by Aleksandrov [[1]]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [[44]] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one-but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are simple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach.

## Keywords

Distinct Vertex Combinatorial Structure Regular Polygon Combinatorial Type Klein Bottle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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