Simple Closed Curve
Date: 05/22/2001 at 00:26:52 From: Robin Subject: Why is it called a simple closed curve? Why is a simple closed curve called just that? I understand the concept that it has no intersecting lines and is any closed shape, but I don't understand what the term "curve" has to do with shapes that have no curve. What "curve" is being used to define shapes such as squares, triangles, etc.? I'm trying to explain this to my 4th grader. Thank you.
Date: 05/22/2001 at 08:47:55 From: Doctor Peterson Subject: Re: Why is it called a simple closed curve? Hi, Robin. You apparently understand "simple" (all one piece, not divided into parts by intersecting itself) and "closed" (connecting back on itself so as not to leave any dangling ends), but have trouble with "curve." I can understand that! This is an example of the way mathematicians like to generalize concepts as broadly as they can, to make definitions simple, while still using basic words that may have more specific connotations in everyday use. A "curve" is anything you could draw with a pencil by moving it around without lifting it; we don't care whether it is straight, or curved, or has sharp bends - even though the word originally meant "not straight" - because we're choosing to ignore those aspects and look only at the most basic feature of a curve. (In the field of math called topology, in which this term is used, we ignore the actual shapes and sizes of figures and consider only how they are connected.) Perhaps we could have found a word in English or Latin that would not imply the wrong ideas, but I can't think of a better term. In a similar way, a "plane" is a generalization of a sheet of paper, ignoring the material it's made of, the existence of edges, and so on. This sort of abstraction is, in my mind, the foundation of math - but it does make the language of math a little peculiar. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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