Do Curved Surfaces Have Faces, Edges, and Vertices?

Having discussed how to count faces, edges, and vertices of polyhedra, and then looked at Euler’s formula that relates them (not only in polyhedra but in graphs on planes and other surfaces), we need to consider a question we have received at least 100 times: are these terms even defined (or defined correctly) for cylinders, cones, and such?

Does a cone have an edge?

Here is a question from 2001 to start us off:

Parts of a Cone

I am a second grade teacher and we are currently teaching a unit on shapes. The question came up as to whether or not a solid cone has any edges. My contention is that the definition of an edge is where two planes intersect, and therefore a cone cannot have an edge. Another teacher says that the curved surface of a cone represents an infinite number of planes, and therefore represents an infinite number of edges. 

I would very much appreciate your response, and don't be afraid to get technical. This is as much to satisfy my own curiosity as to let the kids know the proper answer.

We had previously answered several questions that touched on this, but had not really addressed the issue. I took the question, which was the start of a “career” of specializing in this question:

We get this question from time to time, and can never really give a definite answer. The word "edge" is used in different ways; often people get in trouble by introducing the concept of "edge" in the context of polyhedra (where it does mean the intersection of two flat faces), but then talking about curved surfaces like cones without additional comment.

Here's the definition in the Academic Press Dictionary of Science and Technology:

  1. in graph theory, a member of one of two
     (usually finite) sets of elements that
     determine a graph; i.e., an element of
     the edge set. The other set is called
     the vertex set; each element of the edge
     set is determined by a pair of elements
     of the vertex set...

  2. a straight line that is the intersection
      of two faces of a solid figure.

  3. a boundary of a plane geometric figure.

In the latter sense (which I think is appropriate in discussing a cone, even though the dictionary only mentioned plane figures and not curved surfaces), the cone has one edge. I definitely would not bring in the idea of "an infinite number of edges"; that kind of reasoning generally leads to trouble! I would simply say that we can extend the concept of edge either from the world of polyhedra (definition 2) or from the world of plane geometry (definition 3) to apply to possibly curved boundaries of possibly curved surfaces, as long as we say that we are doing so. This also agrees with definition 1, which likewise does not require straightness (indeed, there is no such concept in graph theory), and which relates to boundaries when we consider planar graphs (as in Euler's polyhedral formula).

In other words, I did not find a proper definition of “edge” that fits this context, but it is reasonable to extend the definition, when we extend the context, to the nearest analogue of a polyhedron edge, which is the intersection of surfaces. But we need to state our definition when we do so — it is not necessarily something that others will have defined the same way.

What definition you use depends on what you are going to do with it. If you are just describing objects, my loose definition is fine. If you are going to prove theorems involving planes and angles, you'll want to restrict yourself to the polygonal definition, but then you won't be asking any questions about cones. I think people often fail to realize that even though we are very particular about definitions in math, those definitions may vary from field to field, as they are adapted to a certain context. That's what I'm trying to do here.

The same questions arise concerning faces and vertices, and it's even harder to decide in those cases.

Mathematicians are accustomed to making up definitions for their particular context like this. But we’ll have more to say about that.

Does a cone have a vertex?

A year later, we got this question along similar lines:

Does a Cone have an Edge? A Vertex?

Our 4th grade math textbook defines a cone as "A solid figure with one circular face and one vertex."  This sounds reasonable until you read the textbook's definitions for face, edge, and vertex. The textbook defines a face as "A flat surface of a solid."  It defines an edge as "A line segment where two faces of a solid meet." It defines a vertex as "A point where two or more edges meet."  

Assuming that these definitions are accurate and that I'm not misinterpreting them, a cone must not have a vertex. If a cone has only one face, then it can't possibly have an edge. Therefore, if it doesn't have an edge, it can't have a vertex.

Again, the book is giving definitions suitable for a polyhedron, but applying them inconsistently to non-polyhedra. I responded again, after referring to that previous answer:

Elementary texts (and high school texts, for that matter) are not always very careful about definitions. The problem really is that the same word can be used with slightly different but related definitions, and we don't always bother to specify how to modify the definitions when we move to a different context. The definitions given are for a polyhedron. When you talk about a cone or cylinder, you have to either use a different set of words, since "edge" and "vertex" as defined don't apply at all, and "face" applies only to one of the two surfaces of a cone; or you have to modify the definitions to allow curved edges and faces. Taking the latter approach, the cone will have two faces, one curved, and one curved edge. I'm not sure I've ever seen such modified definitions actually stated, but I have no trouble allowing them, as long as we state them clearly! What really bothers me is when a book is consistent enough not to call the curved "face" of a cone a face, but doesn't bother to define a word that we _can_ use for it. When they then go ahead and say it does have an edge or a vertex, children are bound to be confused.

The really tricky part here is that the "vertex" of a cone has nothing to do with edges, so it needs a whole new definition; and I can't think of a really good elementary-level definition for what they obviously mean, which is simply a "point." I prefer to use the word "apex" and avoid the problem.

I later discussed the “vertex” of a cone here:

Conical Vertex?

Does a cylinder have edges?

A month later we got yet another question about this, which got a long answer:

Number of Cylinder Edges

My 8-year-old son was asked "how many edges are there on a solid cylinder?" on a recent math examination.

His answer was "2" and it was marked as incorrect.

He truly believes in his answer and has asked for my assistance in researching.

I answered first, again having referred to the previous answer as background:

It depends on how "edge" was defined in his class, which may not agree with his intuitive definition.

Often, an edge is required to be straight, in which case a cylinder has no edges. Unfortunately, elementary texts are not always very careful about definitions, and they can ask questions like this that are really worthless. The only definition of "edge" that would make sense in this context would be the one your son is naturally using (a boundary between smooth surfaces making up an object), which would allow a cylinder to have two edges. Asking this question with the other definition only invites confusion, so I wish they wouldn't ask it. I'd like to hear how they did define the word.

If you are going to use the word “edge” in talking about a cone, you must use a definition that is appropriate, just as I previously commented about needing some word that describes the curved surface of a cone, if you are going to ask about it at all.

What is the purpose of definitions?

But this discussion went further, outside of Ask Dr. Math. Doctor Sarah added this:

David asked the same question in the discussion group geometry-pre-college, where it received these responses:

http://mathforum.org/kb/message.jspa?messageID=1077941

She copied in two responses to the question. The first, by Walter Whiteley, dug into issues related to Euler’s formula, which I discussed last week, and will talk about more next time. I will just quote pieces of it that relate most directly to our present issue of definition:

This is a common issue among elementary teachers, and some elementary text book writers. Basically different sources put down different answers.

The underlying issue is: What is the context? What is the larger mathematics one wants to engage with? Without this, there are too many plausible responses. ...

He talks about convex polyhedra, then about topology, where straightness and flatness are irrelevant, concluding that the intuitive definitions make sense, with caution.

However, some elementary texts and test writers decide they know best and give distinct definitions of 'faces', 'edges', and 'vertices'.   When doing so, there should be some good mathematical reason for doing that. Some set of situations one is trying to make sense of. Simple extrapolation on one basis or another, without investigating the good and bad patterns, is a source of trouble. That, unfortunately, routinely happens in elementary (and some high school) materials.

If faces are 'flat regions' and 'edges' are straight lines, then a cylinder has two faces, no edges, and there is no real purpose in the answer. It does not even help you calculate the surface area!

If faces are regions, and edges are where two faces meet, then a cylinder has three faces and two edges (no vertices). This still does not seem to be a mathematically interesting description.

Some authors even require a face to be a polygon, so that a cylinder has no faces and no edges.

Definitions with no purpose are contrary to the spirit of mathematics, as well as to pedagogy.

I suspect that whatever answer this particular test expected, it is based on a particular discussion in a particular text. I can show you different materials with different answers, but seldom is there a mathematical discussion. Some people have concluded that, as a result, it is simply a bad idea (distracting without learning) to use the words faces, vertices, edges for such objects. I do not quite agree - but the only really useful context I know is the larger topology, and you can see that this takes a larger understanding, something I only learned at graduate school, and only teach in some upper level undergraduate courses (courses most teachers have not taken).  

Odds are this discussion in the source text or materials did NOT give enough context to explain why one would bother with these words for this object.

What is the MATHEMATICS one is trying to do!
That is where one needs to start.

The only purpose I see for using these terms is to be able to talk about an object, not necessarily even to do math with it. But to do that, the terms should have universal definitions, as these do not.

For a similar question from 2004, see:

Do Cones and Cylinders Have Faces or Surfaces?

Can a face be curved?

I’ll close with a question from a 12-year-old in 2008:

Is a Curved Surface a Face?

Is a curved surface a face or not?  Like in a cylinder is the curved surface considered a face? 

Some people tell me that a curved surface is a face and some say it's not.  When I search in Google I also don't get a straight answer.  I just want to find out.  I think a curved surface is not a face.

I took this as a chance to put together a larger perspective on these questions.

As you've discovered, there is no straight answer to this.

In mathematics, we define terms to meet a need.  If something is worth talking about, we give it a name, and define exactly what that name means.  Mathematicians talk about faces, edges, and vertices commonly in the context of polyhedra, where faces are all flat, and therefore are always polygons, and edges are always straight line segments.  We have not found it very useful to extend this idea to other shapes, such as cylinders or cones, which have curves, so we have not made a standard definition for these terms in that context.  If we happen to need to do so, we would give our definitions at the start of our paper, and would use whatever definitions make it easy to talk about what we want to talk about.

In our experience answering random questions with little or no stated context, it quickly becomes clear that context determines what definitions apply, so we can’t answer effectively without that background. We also see that without a universal definition, mathematicians happily just define them ad hoc.

There are several ways we COULD extend the definitions.  We could leave them just as they are, requiring faces to be polygons, and edges to be straight; but then since cylinders and cones have surfaces that are not faces, we need extra terms for those.  "Curved surface" is a reasonable name; probably we would also talk about "curved edges".

Another possibility is to change the definitions to fit curved objects.  We might require a face to be flat, but not necessarily a polygon, so that the circular bases of a cylinder would be faces, but the "curved surface" would not be.  Or, we might call any surface a face.  The question would be, why do we need to use the terms?  Are there theorems that apply only to what we are calling "faces", and not to other surfaces?  That would determine what is the best set of definitions to use.

I have varied on which approach I prefer. That’s largely because there is no particular reason to choose one over the other. That is also why books disagree.

Elementary textbook authors seem to feel a need to have a word for everything, and to be able to apply each word to all the shapes they want to talk about--to be able to answer the question "How many faces does this have" for any object.  So they decide for themselves (possibly without a valid mathematical reason) how they want to define these terms.  As a result, you get books that use different definitions.  I wish they wouldn't do that, because it confuses a lot of children when they look up an answer and find it disagrees with their book or teacher.  The best thing is just not to bother asking the question at all.

The basic rule ought to be “when in doubt, don’t”. I have taught courses in Mathematics for Elementary Teachers several times, and observed that their textbooks did not ask about faces of cylinders; there was no need to do so. This wisdom needs to be passed down to the elementary curricula themselves.

So the answer to your question is: SOME people consider a curved surface to be a face, and others do not.  Those whose opinion matters most, don't have an opinion (or would ask you for the context of your question before attempting an answer).  If you are asking just for yourself, your answer is fine: a cylinder has two flat surfaces and a curved surface, and two curved edges.  If you are answering a question asked on a test, you'll have to find out what your own text says.

That, unfortunately, is the bottom line.

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