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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.

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Since all Riemannian geometries are, locally, approximately flat, I imagine the intuition would be something like "imagine everything from a bug's perspective." By analogy, we often think of the surface of the Earth as being like a plane, because locally, this is a fairly good approximation. (Minus hills, etc.) – Charles Staats Aug 1 '10 at 20:28
This is more or less the plot of The Inverted World written by Christopher Priest. – Kii Jun 9 at 8:26
up vote 72 down vote accepted

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.

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Fascinating! Suffice it to say I did not expect such a nice answer to this question. – Zach Conn Aug 2 '10 at 20:42
I only had a good answer because I was doing hyperbolic geometry in the previous lecture of the summer course I'm teaching. – Charles Siegel Aug 5 '10 at 4:27
This is nice and intuitive, in that the "lines" of this strange Euclidean geometry are given by nice intuitive geodesics on the horosphere. (Similar to how spherical geometry is more intuitive than the Poincaré disk.) – Matt Apr 21 '11 at 1:12
In fact, a plane obviously yields standard n-1 dimensional hyperbolic geometry, and a plane with a little bit of curvature (i.e. a hypercycle) will work too if you're willing to use geodesics as your lines, but when the curvature gets to the critical point (so the surface becomes a horosphere, and the number of points at infinity suddenly drops to one), then you get Euclidean geometry, and with curvature past the critical point (so the surface becomes a sphere) you get spherical geometry. This makes it very intuitive that Euclidean is right at the boundary between hyperbolic and spherical. – Matt Apr 21 '11 at 1:45

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

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Care to explain why this answer is correct? – Qiaochu Yuan Aug 2 '10 at 3:16
Charles Siegel's answer explains it better than I would have. – Doug Chatham Aug 2 '10 at 12:53
While I am glad you pointed to this link, in the future if you are worried about not explaining well, and just want to provide a link, you should consider just leaving a comment. This seems to be the protocol for this. – BBischof Aug 2 '10 at 14:30
Well, now that I have enough rep points to make comments on questions I didn't ask or answer, I'll do that. – Doug Chatham Aug 3 '10 at 14:16

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.

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I found your post very interesting and got Gans "Transformations and geometries"(1969) second hand, the coordinates of a point become the coordinates ${x',y'}$ from the point where the segment ${0,0,1}$ to ${x,y,0}$ cuts an unit circle centered at ${0,0,1}$. If I am correct this means that $x' = \frac {x}{\sqrt{1 + x^2 + y^2}} $ and $y' = \frac {y}{\sqrt{1+ x^2 + y^2}} $ the book is very interesting. 10 pound well spend :) thanks – Willemien May 30 '14 at 17:34

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:

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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 everybode 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.

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