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Wandering around Wikipedia, I came across the idea that if we violate the parallel postulate, we arrive at new, non-Euclidean geometries. Specifically, if you violate it in one direction, you get elliptic geometry, and in the other direction you get hyperbolic geometry.

It's a fascinating idea, but Wikipedia doesn't say a whole lot about it. I've spent a few weeks turning the idea over in my mind, and I now think I understand it. Basically I want to write down how I think it works, and have someone tell me whether I'm correct or not. However, I'm having trouble not making this into a 50-page essay that nobody will ever read!


As best as I can understand it, it's a question of space. Elliptic geometry doesn't have enough of it. Hyperbolic geometry has too much of it. Let me explain...

Euclidean geometry is the geometry of flat space. If you take a flat sheet of paper, cut wedges out of it, and glue the edges together, it forces the paper to curve. If you follow that curve far enough, it naturally closes into a complete sphere.

Sure enough, if two ships set sail from the north pole on different headings, initially the distance between the two ships grows linearly, just like Euclidean geometry would suggest. However, by the time they reach the equator, they are actually sailing parallel to each other, and after that they actually sail towards each other.

(Question: Is elliptic space finite in size? If you travel in a straight line for long enough, do you end up back where you started?)

Basically, as you travel outwards, elliptic geometry has "too little space", compared to what you would expect from Euclidean geometry.

Hyperbolic geometry is harder to think about; the Earth is spherical, but I'm not aware of any simple real-world shape that is hyperbolic. But, logically, if elliptic geometry is the geometry of missing space, hyperbolic geometry ought to be the geometry of too much space.

That is, as you travel outwards form a point, you find too much space around you. I don't know exactly "how much" extra space, but more than you would expect.

(Question: What's the formula for the circumference of a circle in elliptic geometry and in hyperbolic geometry?)

This suggests that if two ships set sail towards each other, provided they start off far enough apart and the angle between them is shallow enough, the "extra space" that keeps materialising as they travel onward potentially means they could actually miss each other - which would explain why you can have more than one parallel line. At some point, the angle becomes sharp enough that the ships' paths do cross, but at any shallower angle, they will miss. (And there are infinity such angles.)

Is any of this correct? Or an I barking up the wrong tree?

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No ... not too bad. Another thing is that in Euklidean geometry the angular sum of triangles is $180^\circ$ whereas in elliptic geometry the excess over $180^\circ$ is proportional to area. Therefore indeed the elliptic plane (even without referring to the Earth surface) has finite area: It is completely covered by eight equilateral rectangular triangles. Hyperbolic is indeed harder to imagine; think of a saddle (or a Pringels chip). - To complete everything, three-dimensional geometry can also be non-Euklidean (but then you can hardly visualize it in your everyday - Euklidean - 3d world). –  Hagen von Eitzen Jan 19 at 11:06
    
Surely there are more than two ways to curve space. –  Karolis Juodelė Jan 19 at 11:08
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Also worth mentioning is that in elliptic geometry, you cannot "order" a line (that is, you can't take a line, assign "left" and "right" along it and say for certain that one point is left of another). Also, a line doesn't divide elliptic space into two. So as far as basic geometric intuition go, hyperbolic and euclidian can be seen as closer together than to elliptic. This is also reflected in axiom systems such as Hilbert's, where as you say, tweaking the parallel postulate just a little bit gives you hyperbolic from euclidian, but the axioms of betweenness have to be scrapped for elliptic. –  Arthur Jan 19 at 11:13

1 Answer 1

Question: Is elliptic space finite in size? If you travel in a straight line for long enough, do you end up back where you started?

It depends on whether you are thinking globally or locally. In the global case, yes, the space is finite in size, however, if a space is locally elliptic it does not necessarily mean that it is globally elliptic (or finite). In response to your second question, it depends on if you are dealing with a perfect sphere or an ellipsoid (i.e. like a water mellon or a hamburger bun). In those cases one orbit of a godesic will not necessarily bring you back to where you started (see Geodesics on an ellipsoid).

What's the formula for the circumference of a circle in elliptic geometry and in hyperbolic geometry?

If we let $C$ be the circumference of a circle and $D$ be the diameter, then....

for eliptic spaces we have $\frac{C}{D} < \pi$ and,

for hyperbolic spaces we have $\frac{C}{D} > \pi$.

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