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? http://mathforum.org/library/drmath/sets/select/dm_cone_edge.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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.
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