Alternative model of Euclidean geometry

Question

I'm planning to teach high-school geometry. As usual, this will be by building from axioms. (The axioms used are AFAICT particular to the book I've been assigned, but they're some combination of Hilbert's, SMSG's, and God knows what.) I'm considering demonstrating that geometry's axioms need not have their usual model by presenting an alternative model of at least a few basic axioms. Can anyone recommend such a model? I'd need it to be accessible to high schoolers (so, for example, not this).

Answer 1

The rational plane $\mathbb{Q}^2$ is a model for Euclid's five axioms, and I would think (hope?) that it is accessible to high-schoolers. Many common geometric constructions don't work as expected for it; for example, here's an excerpt from Explanation and Proof in Mathematics, p.66:

Answer 2

This is many years later, but that paper which you said was not accessible for high schoolers is really very simple, and could easily be explained to high schoolers. There are two variants of it which are even simpler.

You map the entire plane to a disk, and lines become semiellipses with two endpoints at antipodal points on the disk. If two lines are parallel, their corresponding semiellipses will only meet at the disk boundary. That is about all you need to get started, just playing with ellipses and such and showing the basic axioms are met. This is called the "Gans disk." The only challenge may be in explaining that angles aren't what they seem to be in this model, although there is a slight variant which I will explain later that has the benefit of being conformal. But it should look very intuitive; the whole thing looks like you are looking through the plane through some kind of curved lens, which your brain is already partly wired to "decode" and so on.

There is also a much better map to look at which is also conformal. The "real" map of interest here isn't of the Euclidean plane to the disk, but to the surface of a hemisphere; the disk model above is just an orthographic projection from the hemisphere to the disk. In the hemispheric model, lines map to geodesics (great circles, or rather great semicircles), such that when you "flatten" things orthographically you get the aforementioned ellipses. The gnomonic projection from the center of the hemisphere maps back to the original Euclidean plane, so you are basically just projecting the entire plane onto the hemisphere, which is easy to compute, and then dropping the z coordinate to get your Gans disk. So you can explicitly build the map from the Euclidean plane to the disk if you want. You can literally go get a styrofoam hemisphere and some string and thumbtacks and show how the properties are preserved.

But let's say even this is too complex for your particular class. Well, you can take it even one step further into high school intuition, because there already exists a real life model of this, or something close to it, which everyone already knows: a fisheye camera lens. A fisheye lens is already basically doing this, and everyone knows that lines become curved and such. An "ideal" fisheye lens with a 180° field of vision will be doing precisely this, or something very close to it. At this point it should be immediately apparent how this is of course still a model of Euclidean geometry, just being sent through a certain geometric transformation that will still cause parallel lines not to meet and etc. You can print out a 2D grid on a sheet of paper, and hold it up to the fisheye lens, and show that the result is something similar to the ellipsoid model from before.

So you can present this in two ways: the mysterious way with these ellipses that magically satisfy the axioms, and then the intuitive way with this hemispheric projection and fisheye model. This is also a very important model of Euclidean geometry as it is highly relevant to the kinds of spherical geometry that are important to how cameras work, e.g. computer graphics, 3D video games and even your own eyes and visual cortex. It is related to the very important mathematical result that the Euclidean plane embeds into projective 3-space and so on.

As a last little tidbit, there is another simple disk model of the Euclidean plane that is a slight variant of the Gans disk above, but which is built from a stereographic projection of the hemisphere to the disk rather than an orthographic projection. Instead of making the geodesics semiellipses, they instead are circular arcs that intersect the outer disk at two antipodal points. Given two such points, the set of all arcs that one can draw going though those points (with different radii) is the set of all lines parallel to one another at that particular orientation. Geodesics can be drawn using only a standard compass, and since angles are preserved, the "triangles" you draw this way will always have their internal angles magically add up to 180 degrees and etc. This may be of some benefit in explaining how models of Euclidean geometry are lurking in some strange places, although it doesn't have quite the benefit of being so directly "intelligible" to your brain as the previous models as you aren't quite as used to seeing this kind of distortion as you are with a fisheye lens. Still, though, it is worth looking at, and is technically also derived from the hemisphere.

As a last note, probably too advanced for high schoolers but still kind of interesting, you will note all of these have an analogue in terms of the hyperbolic plane. The "Gans disk" is directly analogous to the Beltrami-Klein disk, and the stereographic projection above is analogous to the Poincaré disk, and the hemisphere model is also the same hemisphere model that corresponds to what you'd get if you embed a camera into hyperbolic space. The only difference is that projected geodesics onto the hemisphere are no longer great circles going through two antipodal points, but smaller circles that intersect the hemisphere boundary at a 90° angle. As a result it is easy to see how you can have "extra" parallel lines that don't exist in the Euclidean plane. Also, since we have the same hemispheric model, in theory we should be able to see what the hyperbolic space looks like just from putting a camera in it and modeling the result. What would that look like? Hm...