How Many Edges in a Circle?
Date: 06/15/99 at 18:12:22 From: Dennis Mayhew Subject: How many edges in a circle Dear Doctor, A group of teachers here in Edmonton, Alberta have been debating over the number of edges in a circle. Three answers have been presented: one, none, and infinite. What do you think? We want to know which one we should be teaching. D Mayhew
Date: 06/18/99 at 07:23:23 From: Doctor Pete Subject: Re: How many edges in a circle Hello, Here is what I think. I do not believe there is an "infinite" number of edges in a circle, by any common or logical definition of the word "edge." An edge can be thought of as a boundary - in which case a circle, being a single closed figure, does not have an infinite number of boundaries. However, does a square, having in a sense one continuous boundary, therefore only have one edge? Certainly we do not wish the term "edge" to be so imprecise as to fail to distinguish between a square and, say, a pentagon. So, one might say that an edge is a continuous straight line on the boundary of a closed figure. But the circle, having no such lines, has *no* edges under this definition. This I find somewhat unsatisfactory, but less so than "infinite." More pleasing to my sense of the word, suppose an "edge" is an open, nonempty subset of the boundary of a simple closed curve that is continuously differentiable. Under this definition a circle has one edge - the entire boundary. A semicircle would have two edges: the curved portion, and the diameter. This seems sensible because we wish to distinguish the circle and semicircle by an important property - a semicircle can be said to have two "vertices." Clearly we would like to say that a vertex is a "point," no matter how blunt, so to speak. A regular 1000000-gon may be close to a circle, but it is still not yet a circle. Precisely speaking, a "vertex" is a point at which the boundary fails to be differentiable. A smooth curve, or a straight line, does not have a vertex. But this last thought brings us back to the beginning. Think of a regular polygon of n sides. As n gets larger and larger, this polygon becomes more and more circular; that is, it gets closer and closer to approximating a circle. One might say that as n approaches infinity, we obtain a circle. And hence, it may be argued that a circle is a regular polygon with infinitely many edges. Well, perhaps so. But there are two reasons why this thinking is not satisfactory. First, how long is an edge in such an infinite-sided polygon? It must have zero length, and therefore is not a line but a point. So can we truly call this zero-length segment an edge? The definition of "edge" I gave does not permit an edge to be a single point - it must contain more than one point. Second, this limiting process, when applied to curves in general, implies that any curved figure has an infinite number of edges - for example, the semicircle would also have an infinite number of edges. But for the reasons previously given, we would like to distinguish between a circle and semicircle. So, in short, it would seem that a circle has one "edge." But in fact the definition that permits this to be so is not a commonly perceived definition, as "edge" in common usage implies "straight line." Thus, I feel that the matter is made more precise by simply not using the word "edge" to describe a circle at all. In closing, here is a question: What kind of simple closed figure would have an infinite number of edges, under the definition I provided? The answer, I think, will amuse your teachers. - Doctor Pete, The Math Forum http://mathforum.org/dr.math/
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