The Importance of Geometry Constructions

Date: 12/29/98 at 21:46:10
From: kel
Subject: Geometry constructions

I am doing a report on constructions in geometry. I would like to know 
why constructions are important. I realize that they challenge us to 
use different tools but there must be more to it then that. So I was 
wondering if you could give me more of a reason why constructions are 
so important? 

Thank you very much.

Date: 12/30/98 at 12:26:56
From: Doctor Peterson
Subject: Re: Geometry constructions

Hi, Kel. That's a good question. We tend to teach it out of tradition, 
and forget to think about why it's worth doing!

Certainly learning how to use the tools is useful. Some of the 
techniques are useful in construction (of buildings, furniture, and so 
on), though in fact sometimes there are simpler techniques builders use 
that we forget to teach. But I think the main reason for learning 
constructions is their close connection to axiomatic logic. If you 
haven't heard that term, I'm talking about the whole idea of proofs and 
careful thinking that we often use geometry to teach. 

Euclid, the Greek mathematician who wrote the geometry text used for 
centuries, stated many of his theorems in terms of constructions. His 
axioms are closely related to the tools he used for construction. Just 
as axioms and postulates let us prove everything with a minimum of 
assumptions, a compass and straightedge let us construct everything 
precisely with a minimum of tools. There are no approximations, no 
guesses. So the skills you need to figure out how to construct, say, a 
square without a protractor, are closely related to the thinking skills 
you need to prove theorems about squares.

You may be interested in looking at Euclid's theorems to see for 
yourself how much involves constructions. Here's a Web site by David 
Joyce of Clark University on Euclid's first book, the first proposition 
of which is a construction:

   http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/