Sum of Angles of a Triangle in Non-Euclidean Geometry
Date: 08/01/2001 at 18:54:21 From: Roger Subject: Three-dimensional figures Recently I was trying to teach my son that the sum of all the angles of a triangle is equal to 180 degrees. My son came up with a drawing of a triangle on a surface of a sphere in which the sides, due to the shape of the sphere, looked like an arch instead of a straight line of a normal triangle drawn on a plane surface. When we added the angles of that particular triangle the sum was definitely greater than 180 degrees. Is that possible? Why?
Date: 08/01/2001 at 23:11:47 From: Doctor Peterson Subject: Re: Three-dimensional figures Hi, Roger. The theorem about the sum of angles in a triangle is from PLANE geometry, and depends on the parallel postulate, which says that, given a line and a point, there is exactly one line through the point parallel to the line. If you move to SPHERICAL geometry, this postulate is no longer true, and you have a "non-Euclidean" geometry. (You have to redefine "lines" to be great circles.) Here, the sum of angles is always greater than 180 degrees; in fact, it turns out that the excess over 180 degrees is proportional to the area of the triangle. Try making a triangle containing 1/8 of a sphere, using the equator and the 0 and 90 degree longitude lines; then try some other similar triangles to see that this is true. You can read about this in the Dr. Math archives: Drawing Triangles http://mathforum.org/dr.math/problems/joe6.18.97.html A Triangle with Three Right Angles http://mathforum.org/dr.math/problems/stine.12.1.99.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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