Studying Euclidean geometry using hyperbolic criteria

Question

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.

But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré's Elements. This postulate is the obvious statement that given a line $L$ and a point $p$ not on $L$ there are at least two lines through $p$ that do not meet $L$. Your friend wonders what it would be like if this assertion were replaced with the following: given a line $L$ and a point $p$ not on $L$, there is exactly one line through $p$ that does not meet $L$.

You begin investigating this Euclidean geometry, but you find it utterly impossible to visualize intrinsically. You decide your only hope is to find a model of this geometry within your familiar hyperbolic plane.

What model do you build?

I do not know if there's a satisfying answer to this question, but maybe it's entertaining to try to imagine. For clarity, we Euclidean creatures have built models like the upper-half plane model or the unit-disc model to visualize hyperbolic geometry within a Euclidean domain. I'm wondering what the reverse would be.

Answer 1

Here's another version of Doug Chatham's answer, but with details.

If you lived in Hyperbolic space, then Euclidean geometry would be natural to you as well. The reason is that you can take what is called a horosphere (in the half-space model for us, this is just a hyperplane which is parallel to our limiting hyperplane) and this surface actually has a Euclidean geometry on it!

So unlike for us, where the hyperbolic plane cannot be embedded into Euclidean 3-space, the opposite is true: the Euclidean plane can be embedded into hyperbolic 3-space! So this is analogous to our understanding of spherical geometry. It's no surprise the spherical geometry is slightly different, however, it fits nicely into our Euclidean view of things, because spherical geometry is somewhat contained in three-dimensional geometry because of the embedding.

Answer 2

Look up "horosphere" (for example, in page 90 of the Princeton Companion to Mathematics). Wikipedia describes it on its Horoball page.

Answer 3

An alternative to the horosphere model ...

In "A Euclidean Model for Euclidean Geometry", Adolf Madur discusses a Disk model of the Euclidean plane. (Madur says that David Gans has priority for discussing this model, so I'll call it the "Gans Disk".) The "lines" consist of diameters of the Disk, and half-ellipses that have a diameter as a major axis; the measure of the angle between two "lines" is defined as the traditional measure of the angle between their respective major axes. With an appropriate metric (which I have forgotten, and which is just missing in the document preview linked), we get all of the Euclidean plane crammed into the Disk.

Overlaying the Gans Disk on the Poincaré Disk (or a sub-disk thereof) provides another way for Hyperbolians to study Euclidean geometry. They just have to agree to treat these half-ellipse paths (which I don't think are ellipses to them) as "lines", and to alter their concept of angle measure and length accordingly.

This model might be considerably harder for Hyperbolians to wrap their minds around than the horosphere model, though.

Edit. Since ellipses are projections of tilted circles, we can "lift" the Gans Disk to a "Gans Hemisphere". (This is actually a middle phase in the derivation of the Gans Disk model.) There, the "lines" are great semi-circles, with angles measured via their diameters in the equatorial plane. Not a major refinement of the Gans Disk, but at least the "lines" are naturally-occurring geometric objects, instead of the contrived ellipse-paths. Of course, the metric would need adjustment; off the top of my head, I don't know how much more (or less?) complicated that metric would be.

Answer 4

Although I fully agree that horospheres would likely yield the best model, another thing woth considering is that at small scales, hyperbolic geometry becomes almost Euclidean. So they might think “well, Euclidean geometry is what you get if you imagine a whole universe bundled up to the size of an atom.” Contrary to the horosphere model, this idea would not allow them to do e.g. drawings in Euclidean geometry, unless either they are masters at drawing on an atomic scale, or the curvature of their universe is so low compared to their body size that it is almost Euclidean even at everyday scales. But everybody there should have a pretty intuitive feeling of what not only the Euclidean plane but the Euclidean space would look like, simply from extrapolating the effects they observe as things become smaller.

Answer 5

In a sense there is one model for Euclidean geometry. However, the geometry of the sphere can be studied on spheres with different radii and, thus, different curvature.

For critters who grew up on a hyperbolic plane there is also a parameter that measures the curvature of their world. Some nice visuals about this and technical details can be found here:

http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html

Answer 6

I came from Hyperbiolea and I find it ridiculous that you, earthlings think that imagining "Euclidean geometry" (how silly a choice of a name) is a big deal.

We have carts -- you know -- and we've been studiyng the trace of the weels for thousands of years. If one of the weels follows a straight line then the other weel follows another path called an equidistant. (The distance beeween the weels does not change! Rigid stuff.)

Now imagine that we single out one point on our plane, a holy point, heh heh, and consider all the straight lines through this point and all the the equidistant lines belonging to them. I suppose that you guys can immediately see the following: (1) two points uniquely determine one equidistant. (2) parallelism is a well defined concept considering these lines and this parallelism is Euclidean. (Note that the straight lines through the holy point can also be considered as rquidistants (0 distance))

With these properties we have (with your primitive word) an affine plane.

We developed the concept of Euclidean congruency as well. (I am not going to detail that.)

If you think that our Euclidean concepts are just apossibilty to fooling around for mathematicians then you are very wrong. Here is our biggest engineering problem that we solved based on Euclidean concepts. Thousands of years ago it happened quite freuently that some of the weels flew away if the cart exceeded a certain speed limit -- many casulties! So we had to take the phenomenon (breaking speed) very seriously. But let you find out about the details of this interesting thing.

Answer 7

Charles' Segal's answer gives an answer about how you could do it if you lived in 3 dimensional hyperbolic space. Here's an answer on how you could do it if you lived in a 2-dimensional hyperbolic space. If you use units of $\frac{1}{\sqrt{2}}$ the size, you get half the amount of curvature. Maybe you will find that that is essentially the same thing as doing abstract thinking that the unit is the same size and the curvature is half the amount. Theoretically, you can extend it to zero curvature and even positive curvature.