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### Do Pyramids Really Exist?

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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
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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/
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Associated Topics:
High School Geometry
High School History/Biography
High School Polyhedra

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