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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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    $\begingroup$ 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.) $\endgroup$ Commented Aug 1, 2010 at 20:28
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    $\begingroup$ This is more or less the plot of The Inverted World written by Christopher Priest. $\endgroup$
    – Kii
    Commented Jun 9, 2016 at 8:26
  • $\begingroup$ I wish you do not object to my rewriting of your previous title. $\endgroup$
    – Jean Marie
    Commented Mar 20, 2020 at 10:48

7 Answers 7

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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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  • $\begingroup$ Fascinating! Suffice it to say I did not expect such a nice answer to this question. $\endgroup$
    – Zach Conn
    Commented Aug 2, 2010 at 20:42
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    $\begingroup$ I only had a good answer because I was doing hyperbolic geometry in the previous lecture of the summer course I'm teaching. $\endgroup$ Commented Aug 5, 2010 at 4:27
  • $\begingroup$ 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.) $\endgroup$
    – Matt
    Commented Apr 21, 2011 at 1:12
  • $\begingroup$ 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. $\endgroup$
    – Matt
    Commented Apr 21, 2011 at 1:45
  • $\begingroup$ Really nice realization! I didn't know that horospheres made up Euclidean spaces before. In that way, you can even have a space, divided by a Euclidean plane, in which the part of the space on one side of the plane is a hyperbolic space and the part of the space on the other side is Euclidean, or also a hyperbolic space, but with another curvature! $\endgroup$ Commented Jan 8, 2023 at 5:26
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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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    $\begingroup$ Care to explain why this answer is correct? $\endgroup$ Commented Aug 2, 2010 at 3:16
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    $\begingroup$ Charles Siegel's answer explains it better than I would have. $\endgroup$ Commented Aug 2, 2010 at 12:53
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    $\begingroup$ 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. $\endgroup$
    – BBischof
    Commented Aug 2, 2010 at 14:30
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    $\begingroup$ Well, now that I have enough rep points to make comments on questions I didn't ask or answer, I'll do that. $\endgroup$ Commented Aug 3, 2010 at 14:16
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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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    $\begingroup$ 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 $\endgroup$
    – Willemien
    Commented May 30, 2014 at 17:34
  • $\begingroup$ Are you sure that lines manifest as half ellipses in this model? This seems a bit odd to me, for example by considering that two half ellipses can be tangent to each other with the tangent point lying inside of the boundary. The fact that the two half ellipses tangent each other at that point indicates that the hyperbolic lines they represent are parallel and go through the same point, meaning that they should be the same line, which they are clearly not since they are represented by two different half ellipses. Or did I misunderstand you? $\endgroup$ Commented Dec 25, 2021 at 4:30
  • $\begingroup$ @HelloGoodbye: "Are you sure that lines manifest as half ellipses in this model?" Specifically, a half-ellipse that has one of the boundary circle's diameters as its major axis. No two of those are tangent inside the boundary circle. ("Parallel" half-ellipses are tangent to each other at diametrically-opposite points on the boundary circle, but those are "points at infinity", so that's okay.) $\endgroup$
    – Blue
    Commented Dec 25, 2021 at 5:36
  • $\begingroup$ Could you please describe how this Gans disk model works? What are some of its properties? So far it's shrouded in mystery, and I don't seem to be able to find very much inforation about it on the web. And you're not simply talking about the Gans model, are you? $\endgroup$ Commented Dec 27, 2021 at 20:47
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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 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.

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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

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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.

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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.

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