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/