Number of Cylinder Edges

Date: 04/01/2002 at 12:16:20
From: David Saray
Subject: Number of edges in a solid cylinder

Dear Sir:

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.

Can you assist?
Thank you.

Date: 04/01/2002 at 12:33:11
From: Doctor Peterson
Subject: Re: Number of edges in a solid cylinder

Hi, David.

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.

You can read some other discussions of related issues here:

   Does a Cone have an Edge? A Vertex?
   http://mathforum.org/dr.math/problems/mark.03.12.02.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 04/01/2002 at 13:03:33
From: Doctor Sarah
Subject: Re: Number of edges in a solid cylinder

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   

Subject: Re: How Many Edges On  SOLID CYLINDER
Author:  Walter Whiteley <whiteley@mathstat.yorku.ca>
Date:    Mon, 1 Apr 2002 12:48:47 -0500 (EST)

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.

One is almost certainly starting with the known definitions and 
answers for convex polyhedra. The faces are flat plane regions. The 
edges are where two faces meet (and lie along lines). The vertices are 
where three or more faces meet (three or more edges as well) and are 
points. The overall pattern has a nice mathematical structure given by 
Euler's formula  |V| - |E| + |F| = 2.

Now one wants to extend this to other creatures. Perhaps non-convex
but spherical polyhedra. Perhaps general topological surfaces (the 
counts do not change if the polyhedron is made of rubber and deformed 
without cutting or gluing). Perhaps higher dimensions.

If the context is topology, then one works down the topological 
features of the convex polyhedron.
  (a) A face is a topological polygon - a disc with boundary of edges
      and vertices.   
  (b) An edge is a closed curve, with two boundary edges - vertices.
  (c) A vertex is a point.   

Together the vertices and edges form a connected graph. (Whether you 
allow two edges between a fixed pair of vertices, or a loop which has 
the same vertex at both ends is not critical for the topology.)

With this in mind, you still can talk about spherical topology. 
Basically, a connected graph drawn on a sphere without crossings, with 
the regions cut out forming the faces. This still satisfies Euler's 
formula: |V| - |E| + |F| = 2.

If you draw a graph on, say, a torus, or make a torus out of polyhedron
pieces, so that the faces are still discs, then you have a new form
of Euler's formula  |V| - |E| + |F| = 0.  Similar formulae exist for 
covering other surfaces with discs, edges, vertices.

However, to make those formulae work in that simple context, you need 
to ensure the faces have a single polygon as the boundary, and the 
edges do have vertices at their ends.

How might this relate to a cylinder? Well, in many elementary texts,
one studies the 'net' - the flat paper pieces which one might use to 
fold up to the surface.  The standard net for a cylinder has two 
circular discs on the ends, and a rectangle which is to be taped 
together along two opposite sides to form the sides of the cylinder.  
If you study the taping of the net, you can tape it up with three 
pieces of tape (one at each end, and one along the sides). You have 
two points at which a couple of pieces of tape meet - the ends
of the side slit. You have three faces which are discs. In this image, 
you have  |V|=2, |E|=3, |F|=3  and |V|-|E| + |F| = 2. So this image 
makes good sense from the point of view of topology and counting with 
Euler.  

Note that we had to slit the tube of the cylinder, creating an extra
edge, in order to make that face a disc, and to restore the formula.
It is, however, a sensible process.

In the same spirit, one would have to cut up a sphere in order to make 
it work in the topology. E.g. put down an equator, with a vertex where 
the two ends of the equator meet. This would give
|V| - |E| + |F| = 1 - 1 + 2 = 2.

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 not 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.
  
And it still does not really help with calculating surface area - you 
need to cut it open as a net and make the tube into a rectangle. Then 
you have formulae for the areas of faces, and you also happen to have 
the three edges, three faces, and two vertices needed for the pattern 
of Euler's formula. You still see, in the simple count, what the 
overall topology is (the 2 tells you it is spherical and could be 
redrawn, topologically, on a sphere).  

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.

Walter Whiteley

[P.S.  I write this in part because I am trying to have this 
conversation with some textbook publishers for elementary materials, 
and with some other curriculum sources for our schools. It IS an 
important topic to make sense out of, and I look forward to other 
contributions on the topic. Are there other ways to make mathematical 
sense out of the choices?]

-----------------------------

Subject: Re: How Many Edges On  SOLID CYLINDER
Author:  Guy Brandenburg <gfbranden@earthlink.net>
Date:    Mon, 01 Apr 2002 17:47:21 -0500

I suppose it depends on what you define as an edge. I believe that 
most math types would think of it as a straight line segment that is 
the boundary of two polygonal faces; it runs between two vertices, 
which are points where 3 polygonal faces meet. Now, since a cylinder 
has no polygonal faces, then it has no edges at all. 

That being said, many people not familiar with mathematical 
terminology might indeed consider the boundaries of the two circles as 
being edges. But that would not correspond to the normal mathematical 
definitions of polyhedra, prisms, pyramids, and so on. A cone is not 
the same thing as a pyramid, for example; a cylinder is not a prism.

On the other hand, one could think of the circular bases as being
polygons with an infinite number of sides. In that case, there would 
be an infinite number of edges. But that would make the lateral, 
curved, sides into no longer curved sides but an infinite number of 
infinitely thin rectangles. Probably better not take that approach 
unless you are trying to do calculus-type computations.

I am guessing that the expected answer was zero.

Guy Brandenburg

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/