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Topic: How Many Edges On SOLID CYLINDER
Replies: 3   Last Post: Oct 26, 2004 9:16 AM

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Walter Whiteley

Posts: 418
Registered: 12/3/04
Re: How Many Edges On SOLID CYLINDER
Posted: Apr 1, 2002 12:48 PM
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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 to many plausible responses.

One is almost certainly starting with the know 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). Pehaps higher dimensions.

If the context is topology, then one works down the topological
features of the convext 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 polydron
pieces, so that the faces are still discs, then you have a new form
of Euler's formula |V| - |E| + |F| =0.
Similar foruma exist for covering other surfaces with discs, edges, vertices.

However, to make those forumla 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. I 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, with out 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. 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).
Still does not seem to be a mathematically interesting description.

Still does not really help with calculating surface area
- 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 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 base 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 is 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

[ps I write this in part because I am trying to have this
converstation with some text book 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?]

> I was amazed t the previous email chain.
> I too am trying to assist my grader research an incorrect test answer
> that he thinks was correct.
> How many edges are there on a solid cylinder?
> He put "2" he got it wrong and asked his dear old dad to research
> onthe net and find if it was really wrong.
> Thanks!

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