Non-Euclidean Geometry for 9th Graders
Date: 23 Dec 1994 13:10:38 -0500 From: Cerreta, Pat Subject: Non-Euclidean geometry Dear Dr. Math, I would to know if there is non-euclidean geometry that would be appropriate in difficulty for ninth graders to study. Thanks for your patience. Pat Cerreta at Byram Hills
Date: 2 Jan 1995 13:54:00 -0500 From: Dr. Ken Subject: Re: Non-Euclidean geometry Hello there! Euclidean geometry is the kind of geometry that assumes the Euclidean parallel postulate. This states that given any line and any point not on that line, there is exactly one line through that point which is parallel to the given line. It is important to remember that "parallel" always means "lines that never intersect", i.e. "lines that share no common point", NOT "lines that are the same distance apart everywhere." Well, non-euclidean geometry is the geometry that you get when you assume the negation of the Euclidean parallel postulate: there is some line and some point on that line such that either (a) there is no line through that point which is parallel to the given line, or (b) there is more than one line through that point which is parallel to the given point. As it turns out, you can show that if case (a) happens somewhere in your geometry, then there are no parallel lines _anywhere_ in your _whole_geometry_. This is pretty amazing. It means that every pair of lines in this geometry will intersect somewhere. This kind of geometry is called "Spherical" or "Elliptical" geometry. You can model it on the surface of a sphere, which I'll talk about later. As a consequence, you can show that if case (b) happens anywhere in your geometry, then it happens _everywhere_, too. We call this geometry "Hyperbolic" geometry. This is geometry in which, given any line and any point not on that line, there is more than one line through that point which is parallel to the given line. As it turns out, you can then show that there are _infinitely_many_ such parallel lines through your given point and your given line. Anyway, here is a model for Elliptical geometry. Take the upper half of a sphere as the plane you're working in, and let lines in this model be arcs of great circles on the sphere (great circles are circles [on the sphere] whose plane contains the center of the sphere. Examples on Earth include the equator and the meridians, but not the tropic of cancer or the arctic circle). Points are just normal points. Then any two "lines" you draw are going to intersect in some point. Actually, it's a little trickier than that. See, there's a problem at the boundary of the hemisphere. You have to include only half of the boundary, because you don't want lines to intersect in two different points, and if you included the whole boundary, some would. If you included none of the boundary, you'd have some lines that don't intersect anywhere (basically, you want to start out with a whole sphere and get rid of points until there are no antipodal points left - antipodal points are points directly across from each other on the sphere, like the north pole and the south pole). So that's Elliptical geometry. Here's a model for Hyperbolic geometry. Let the plane you're working in be the upper half of the x-y plane, without the x-axis. In other words, the set of all points where y>0. We call this the open upper half-plane. There are two kinds of lines in this model. One kind will be half-circles whose center is on the x-axis, and the other kind will be lines that are perpendicular to the x-axis. Points, again, are just normal points in the plane. So draw a couple of "lines" in this model, and note that at best they'll intersect at only one point. Then pick one of the lines you've drawn and take some point not on that line, and see how many lines you can draw through that point that are parallel to (don't intersect) the given line. A bunch, right? So this is Hyperbolic geometry. I hope you can do some neat things with these models. There's a great book that deals with this stuff. It's called "Euclidean and Non-Euclidean Geometry", and it's by a guy named Greenberg. I forget his first name. The book may be a little advanced for ninth graders to use as a textbook (I used it in college), but maybe you could get some ideas from the book to show them, anywyay. There is also a great deal of wonderfully interesting historical material on the development of this stuff in there. Anyway, I hope this helps you out! Fortunately, our school is on break now, so I had plenty of time to write your response. Write back if you have more questions! -Ken "Dr." Math
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