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Projective geometry.

Spheres and infinite planes: ironing out the wrinkles

You may have noticed that the plane and the sphere don't perfectly match up. First of all, the "opposite" infinite points on the plane (which are really the same point) seem to correspond to two genuinely different points on the equator of the sphere. This is no good. Second, no point on the plane corresponds to the many points on the top half of the sphere. The line of sight from them goes "skyward." Each of these presents us with a problem matching up our infinite plane with our sphere diagram. How can we resolve them?

In order to resolve the problem of the two points which are really the same point, we cannot allow two opposing points across the equator from one another to be "different." We must merge these points into one. This suggests what we should do about the points on the top of the sphere as well. We merge them with the opposing points on the bottom of the sphere, so that each point gets merged with exactly one point, on the exact opposite of the sphere.

What are we left with, after performing this strange operation on the sphere? Certainly not a sphere, but another kind of shape which is exceedingly hard to draw. (As mathematicians say, it "does not immerse into 3-space.") This shape is called the "projective plane," and is identical to the infinite plane we've been discussing before. In fact, it satisfies the axioms of geometry which we desire, so long as we're careful to interpret "point" to mean "pair of opposing points on the sphere," and "line" as "great circle on the sphere:"

1. Through every two points (on the sphere), there passes exactly one great circle (=line).
2. Every two lines (great circles on the sphere), there is exactly one point (exactly two points, opposite the sphere).

In other words, it's a valid geometry in which there are no parallel lines. Obviously, this geometry is different from our regular geometry. It's known as "NonEuclidean geometry." Specifically, it's known as Projective geometry, or elliptical geometry. There is widespread suspicion (among physicists and astronomers) that elliptical geometry is closer than Euclidean geometry to the geometry of our universe. Of course, our universe has four dimensions instead of two!

Puzzles to try

1. Find a triangle in Projective geometry whose angle measures are 90, 90, and 90 degrees, as measured on the sphere. No, that was not a typo. <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">
2. What does your triangle look like when projected from the sphere to the plane? <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">
3. Can you find a triangle in projective geometry whose angle sum (angles measured on the sphere again) is 180 degrees? 360 degrees? <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">
4. Draw a small circle on the sphere in such a way that it crosses the equator twice. Of course, draw the opposing circle on the other side of the sphere. When matched up to the plane, what does this shape look like? Hint: It hits two "infinite points." <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">

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The page last modified Thursday, 13-May-2004 15:03:01 EDT