In working on a proposal for a book about three-dimensional polyhedra last week, I needed to write a brief section on polygons. I found that there were so many different types of polygons with such interesting properties, I thought it worthwhile to spend a day talking about them. If you’ve never thought a lot about polygons before, you might be surprised how much there is to say about them….
So start by imagining a polygon — make it a pentagon, to be specific. Try to imagine as many different types of pentagons as you can.
How many did you come up with? I stopped counting when I reached 20….
Likely one of the first to come to mind was the regular pentagon — five equal sides at angles of 108° from each other. Question: did you only think of the vertices and edges, or did you include the interior as well?
Why consider this question? An important geometrical concept is that of convexity. A convex polygon property has the property that a line segment joining any two points in the polygon lies entirely within the polygon.
The two polygons on the left are convex, while the two on the right are not. But note that for this definition to make any sense at all, a polygon must include all of its interior points.
Convex polygons have various properties. For example, if you take the vertices of a convex polygon and imagine stretching a rubber band beyond the vertices and letting it snap back, the rubber band will describe the edges of the polygon. See this Wikipedia article on convex polygons for more properties of convex polygons.
Did the edges of any of your pentagons cross each other, like the one on the left below?
In this picture, we indicate vertices with dots to illustrate that this is in fact a pentagon. The points where the edges cross are not considered vertices of the polygon. The polygon on the right is actually a nonconvex decagon, even though it bears a resemblance to the pentagon on the left.
But not so fast! If you ask Mathematica to draw the polygon on the left with the five vertices in the order they are traversed when drawing the edges, here is what you get:
So what’s going on here? Why is the pentagon empty in the middle? When I gave the same instructions using Tikz in LaTeX (which is how I created the light blue pentagram shown above), the middle pentagon was filled in.
Some computer graphics programs use the even-odd rule when drawing self-intersecting polygons. This may be thought of in a few ways. First, if you imagine drawing a segment from a point in the interior pentagon to a point outside, you have to cross two edges of the pentagon, as shown above. If you draw a segment from a point in one of the light red regions to a point outside, you only need to cross one edge. Points which require crossing an even number of edges are not considered as being interior to the polygon.
Said another way, if you imagine drawing the pentagram, you will notice that you are actually going around the interior pentagon twice. Any region traversed twice (or an even number of times) is not considered interior to the polygon.
Why would you want to color a polygon in this way? There are mathematical reasons, but if you watch this video by Vi Hart all the way through, you’ll see some compelling visual evidence why you might want to do this.
We call polygons whose edges intersect each other self-intersecting or crossed polygons. And as you’ve seen, including the interiors can be done in one of two different ways.
But wait! What about this polygon? Can you really have a polygon where a vertex actually lies on one of the edges?
Again, it all depends on the context. I think you’re beginning to see that the question “What is a pentagon?” is actually a subtle question. There are many features a pentagon might have which you likely would not have encountered in a typical high school geometry course, but which still merit some thought.
Up to now, we’ve just considered a polygon as a two-dimensional geometrical object. What changes when you jump up to three dimensions?
Again, it all depends on your definition. You might insist that a polygon must lie in a plane, but….
It is possible to specify a polygon by a list of points in three dimensions — just connect the points one by one, and you’ve got a polygon! Of course with this definition, many things are possible — maybe you can repeat points, and maybe the points do not all lie in the same plane.
An interesting example of such a polygon is shown below, outlined in black.
It is called a Petrie polygon after the mathematician who first described it. In this case, it is a hexagon — think of holding a cube by two opposite corners, and form a hexagon by the six edges which your fingers are not touching.
There is a Petrie polygon for every Platonic solid, and may be defined as follows: it is a closed path of connected edges such that no three consecutive edges belong to the same face. If you look at the figure above, you’ll find this is an alternative way to define a Petrie hexagon on a cube.
And if that isn’t enough, it is possible to define a polygon with an infinite number of sides! Just imagine the following jagged segment continuing infinitely in both directions.
This is called an apeirogon, and may be used to study the tiling of the plane by squares, four meeting at each vertex of the tiling.
And we haven’t even begun to look at polygons in other geometries — spherical geometry, projective geometry, inversive geometry….
Suffice it to say that the world of polygons is much more than just doodling a few triangles, squares or pentagons. It is always amazes me how such a simple idea — polygon — can be the source of such seemingly endless investigation! And serve as another illustration of the seemingly infinite diversity within the universe of Geometry….