Do Pyramids Really Exist?

Date: 02/27/2002 at 15:27:01
From: Joeli
Subject: Do Pyramids really exist?

Hi, 

I was just wondering if certain shapes can really exist. For example, 
if the base of an isosceles triangle is 4, and the height is 5, then 
using the Pythagorean theorem the sides are equal to the square root 
of 21. How can this triangle exist (except in theory) if you can never 
measure or draw the square root of 21? I have yet to find a triangle 
that does not involve square roots. Is there one? 

What about pi? Do circles, cylinders, cones not exist because you 
can't measure a distance of pi? 

Thanks. 
Joeli

Date: 02/27/2002 at 16:13:24
From: Doctor Rick
Subject: Re: Do Pyramids really exist?

Hi, Joeli.

You're asking very good questions, the same questions the ancient 
Greek mathematicians asked. They started out assuming that, if you 
took any two lines, you'd be able to find a line (maybe very short, 
but finite) such that the length of each of your lines was a multiple 
of the length of that same short line. It came as a shock to them to 
discover that not all pairs of lines are "commensurable": the ratio of 
the hypotenuse of an isosceles right triangle to a leg is the square 
root of 2, and this is not rational. 

It took some time for the Greek mathematicians to recover from this 
shock. It is probably part of the reason that they focused so much on 
geometry rather than algebra; they distrusted numbers, since they 
could not write the square root of 2 as a number; but they could 
construct an isosceles right triangle with a compass and straight-
edge, so they could represent the square root of 2 geometrically.

Do you notice that you and the Greeks are seeing this from opposite 
viewpoints? You are questioning whether these geometrical shapes 
can "exist" because the lengths are incommensurable; the Greek 
mathematicians held onto geometrical "reality" but had trouble with 
numbers because of the incommensurability problem.

The Greek mathematician Eudoxus apparently came up with the resolution 
that allowed them to go on with their mathematics in the face of this 
problem. You can find Eudoxus' biography at the MacTutor History of 
Mathematics archive:

   http://www-groups.dcs.st-and.ac.uk/~history/Indexes/E.html   

Euclid's presentation of Eudoxus' approach of the incommensurability 
problem is found in Book V of his Elements:

   Euclid's Elements (David Joyce)
   http://aleph0.clarku.edu/~djoyce/java/elements/bookV/bookV.html   

By the way, there are triangles whose sides are commensurable. You can 
just pick any 3 integers and make a triangle with these sides. 
Moreover, there are right triangles whose sides are commensurable: 
the 3-4-5 right triangle is most well known. See Pythagorean triples 
from the Dr. Math FAQ: these are sets of integers that are the sides 
of right triangles.

   http://mathforum.org/dr.math/faq/faq.pythag.triples.html   

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