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Is a Curved Surface a Face?

Date: 03/19/2008 at 20:50:53
From: Abhishek
Subject: face or curved surface

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.

Date: 03/19/2008 at 22:22:45
From: Doctor Peterson
Subject: Re: face or curved surface

Hi, Abhishek.

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.

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.

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.

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.

See this page for some more thoughts:

  Selected Answers: Cone, Cylinder Edges? 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum

Date: 03/19/2008 at 23:02:17
From: Abhishek
Subject: Thank you (face or curved surface)

Thank you for the feedback.  I wanted to know for myself and to figure
out the answer on the test I took yesterday.  Thank you again.
Associated Topics:
High School Higher-Dimensional Geometry
Middle School Higher-Dimensional Geometry

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