Living on a Projective Plane

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Living on a Projective Plane

Postby mitchellmckain on November 1st, 2016, 5:48 pm 

Ever heard of a mobius strip? (you have probably heard of this one)
How about a Klein bottle? (maybe, maybe not?)
How about a projective plane? (much less likely)

The latter 2 are continuous surfaces which are very hard to draw because they don't fit inside a 3d space any better than a map of the Earth fits on a flat piece of paper. There are distortions and discontinuities. It is last of these which is the topic of this post.

There are several ways of constructing these projective planes.
1. You can make an infinite projective plane by adding to a regular infinite plane a single point at infinity in all directions.
2. You can make a finite projective plane by identifying the opposite points of a sphere as the same point.
3. You can also represent a finite projective plane by a circular map by connecting the opposite points on the edge. You walk off the map at one point and you come back on the map at the opposite point. This is nothing like a map of the Earth, right? (at least I hope you get that) You can do it with a rectangular map, but I think I would have to show you so it is done right.

What is like to live on finite projective plane rather than on a sphere? Well the most important consequence is there is only a north pole and no south pole at all.

So why am I contemplating such weird geometry today? There are creatures in my third book who live on a projective plane. They see the stars of our universe in the sky but instead of moving one direction as the planet turns they see two different sets of stars moving past each other in opposite directions, all going in a circle around the north pole. It probably helps to imagine that there is north star occupying that point.

Ok, so what happens if you start at the north pole and walk away from it? You eventually reach a point where you cannot see the north star and half the stars are moving east and the other half are moving west. This is the farthest point from the north pole. Keep walking in the same direction and the north star appears in the sky on the horizon in front of you and you eventually find yourself back at the north pole again.

The projective plane is hard to visualize. It is harder to make picture of than a Klein bottle. google it and you can see the attempts. You couldn't have a planet or asteroid with projective plane surface. The reason one is in my book because it is simulated world. You certainly can make a game map which is a projective plane. But like the Klein bottle you can't really have one in a world of only 3 dimensions.

Image

The point at the center is the "north pole." The crease is because the lines have to cross over (kind of like a mobius strip).

It is convenient to identify the point at the center of this picture above as the north pole, but there is no reason why it must be. In my novel the only thing that makes the north pole different from any other point in their world is the movement of stars in the sky around "the north star." This would include the same peculiarities we see in the day night cycle at the north pole of the Earth.

In the picture it would seem that standing on the north pole would be weird because there is so many directions to go. But this actually not the case. It is no different than standing on our own north pole. So what is going on in the picture?

What looks like a 360 degree angle at the north pole in the picture is actually only 180 degree angle for those actually living in the projective plain. But notice how it crosses over at the crease. that is where the other 180 degrees are in the picture -- on the bottom half. So while the picture makes it look like you can go either up or down from the north pole at a particular angle these are actually not the same angle but 180 degrees apart (i.e. in opposite directions from the north pole). Notice this agrees with my description of a walk away from the north pole where you end up coming back to the north pole from the opposite direction.

Is there anything peculiar going on at the crease? No. There is no actual intersection on the crease. There are two separate not connected points: one for the line crossing down from the upper half to the lower and one for the line crossing up from the lower half to the upper. There is only a crease because of the difficulty of representing this topology in a 3 dimensional euclidean space. Where you put that crease in the picture is completely arbitrary and makes no difference.

There is a peculiarity at the equator where a walk around the equator requires only half the time to get back to the exact same point as a walk off the equator. In other words if you walk parallel to the equator only two feet away from it, then you will come to a point which is only four feet from where you started and only by continuing the journey and going twice as far will you arrive back at the exact same place you started.

The differences between the sphere and projective plane are interesting.

Sphere: lines which start out with parallel tangent vectors intersect at two points on the sphere.
Projective Plane: lines which start out with parallel tangent vectors intersect at one point on the Projective plane.

Sphere: All the great circles through a point intersect at another point which you call the antipodal point.
Projective Plane: All the great circles through a point only intersect at that point. So there is no antipodal point.

Sphere: For each great circle there are two points equidistant to all the points on the circle.
Projective Plane: For each great circle there is only one point equidistant to all the points on the circle.

So if you take a great circle through a point on a projective plane, how do you get to the point which is equidistant to all the points on that great circle?
answer: You make another great circle at 90 degrees to the first one and go halfway around that circle and that is the point.

For the sphere you would also make another great circle at 90 degrees but you would only go a quarter of the way around the 2nd circle in either direction to get the two points equidistant to all the points of the first great circle.
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Re: 4D Inflection

Postby Faradave on November 2nd, 2016, 1:24 am 

Interesting.

I'm more familiar with projection in a Riemannian geometry of a plane onto a sphere, where the sphere sits at an arbitrary origin on an infinite plane, and the points in the plane map on to the sphere. The plane's origin maps to the sphere's south pole and all the other points of the sphere connect to a corresponding point in the plane by a tangent line. The upper pole of the sphere, corresponds to infinity in every direction. ( tangents to the upper pole are parallel to the plane, so by definition, they meet the plane at infinity!) Thus, positive and negative infinities connect at the same pole.

Riemann projection.png
Infinite points on a number line have a 1:1 correspondence to locations on a circle. Shown as a 1D projection, this concept may be dimensionally extended by rotation about the polar axis.

A simple construction is to imagine a flat number line, curved into a circle with the two infinite ends meeting opposite the line's origin. With a true Riemannian projection however, the density of integers varies from the uniformity of the number line, increasing asymptotically toward the upper pole.

They say a Klein bottle is equivalent to attaching the edges of two Möbius strips. I wonder if your projective plane can be constructed from a simple element? Perhaps a rotation of a Möbius strip.
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Re: Living on a Projective Plane

Postby hyksos on May 15th, 2017, 2:41 am 

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