A haven’t made a post in quite some time in my “What is a Geometry?” thread. In working on my online lectures in the section on Polygons, I of course needed to define just what a polygon is. This turned out to be a little more challenging than I had imagined. I thought that the issues that arose would make this discussion an interesting continuation of the “What is a Geometry?” thread.
In general, I think that the Wikipedia does a good job with mathematics — but specifically, the definition of a polygon leaves quite a bit to be desired. I’ll reproduce it here for you:
In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain, or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon’s vertices (singular: vertex) or corners.
OK, maybe not so bad of a start. There are lots of examples given which fit this definition, but many which do not. For example, this definition allows consecutive segments to lie on the same line, which is typically disallowed in most other definitions of polygons.
So maybe a clause may be added to the definition which does not allow this. But then we encounter a polygon like this (I’m using screenshots from my lecture as illustrations):
My definition begins with a list of vertices — but the problem is still the same. The vertex labeled “4” is on the edge joining the vertices labeled “1” and “2.” Again, this is usually avoided.
And what about the following figure?
With the Wikipedia definition, a vertex can be an endpoint of more than two edges of a polygon. Again, problematic. There would be no way to distinguish this figure from a single polygon and two different triangles sharing a vertex.
Moreover, there is no condition saying that the straight line segments need to be distinct. So the same segment might occur multiple times as an edge of a polygon.
None of these behaviors is illustrated anywhere on the Wikipedia page. I’ve done some Wikipedia editing a while back, and would be interested in working on this page when I have more time to devote to such things.
So what is the fix? I’m using the Geometry text of the University of Chicago School Mathematics Project as a reference, which is one of the most rigorous geometry texts around. Here is their definition:
They remark that this is the definition used in 23 out of the 45 geometry text they surveyed. And in fact, it is the rewrite of a definition in previous editions: “A polygon is the union of segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.” This definition was problematic, though, since by this definition, the following is actually a polygon!
Now this revised definition solves all of the problems above — but I couldn’t use it. Why not?
One of the sections I’ll be writing lectures for is three-dimensional geometry — and (of course) I’ll be saying a lot about polyhedra in this section. There are Platonic and Archimedean solids, as well as the Kepler-Poinsot polyhedra, like the small stellated dodecahedron shown below.
The faces of the small stellated dodecahedron are pentagrams, five meeting at each vertex.
But the UCSMP definition does not allow edges to cross. Each edge meets exactly two others, at each of its endpoints. So that means that an edge cannot cross another in its interior.
Now I just can’t talk about polyhedra without talking about nonconvex examples. Sure, it is possible to talk about pentagrams with edges crossing as decagons without crossing edges.
But this would be the height of absurdity. Besides the fact that none of the dozens of books and articles I’ve read on polyhedra in the past few decades ever do such a thing — and I’m sure none ever will.
So I had to go it alone. I’ll share with you my definition — but I can’t say it’s the best. The difficulty lies with being mathematically precise while still making the definition accessible to high school students. Here it is:
A polygon is determined by a list of its vertices. Edges of the polygon connect adjacent vertices in the list, and there is also an edge connecting the last vertex in the list to the first one. All vertices in the list must be different. Finally, no three consecutive vertices of the polygon can lie on the same line, and no vertex can lie in the interior of another edge.
I don’t think this is too bad. But there is still a subtle glitch, which I haven’t worked out yet, and which doesn’t necessarily need to be worked out at this level. When I talk about triangles, for example, I allow cases where the sides have lengths 3, 4, and 7, for example. But I qualify such a triangle by calling it a degenerate triangle.
Since a triangle is a polygon, a degenerate triangle should be a degenerate polygon, right? The problem is that calling something a “degenerate polygon” gives the impression that it is actually some type of polygon. But a degenerate triangle, by my definition, is not a polygon. So when I use the term degenerate polygon, I’m not actually talking about a polygon….
So I’ll let you think this over. I just wanted to share how surprised I was at how subtle the definition of something so “simple” could be. An ordinary polygon.
If you find this sort of question intriguing, you might go online and research all the various definitions of polyhedron. Convex polyhedra are easy to define (as are convex polygons), but when you get into the different types of behavior possible in the nonconvex cases, well, it becomes problematic. In fact, no one, as far as I know, has ever come up with a satisfactory definition for “polyhedron.” Might even do a blog post on that some day….