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/
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