Alternative model of Euclidean geometry 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).
 A: 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:



A: The rational plane $\mathbb{Q}^2$ does not model Euclidean geometry.  You can see this because in Euclidean geometry the diagonal of a unit square has irrational length.  I'll go back to the historical and logical issues at the end.
The interesting alternative model of Euclidean geometry takes "constructible numbers" as coordinates.  You get these by starting with the rational numbers and adding a square root for every positive number.  Formally you suppose for every number $x$ there is also a square root $\sqrt{1+x^2}$.
Cantor and Dedekind in the 19th century already knew  this a countable model, which means it is not continuous anywhere.  As the rational numbers sit as infinitely dense but nowhere-connected dust in the real line, so do the constructible numbers.  This model sits as infinitely dense nowhere-connected dust in the real plane.  
And there remain hard logical questions about this model.  See https://mathoverflow.net/questions/142395/is-the-field-of-constructible-numbers-known-to-be-decidable 
Many historians claim that $\mathbb{Q}^2$ does model the axioms Euclid actually gave, and so they conclude that Euclid's own axioms are defective and cannot even prove Proposition 1.1 constructing an equilateral triangle.  The logical flaw in Euclid's proof (on this reading) is that his axioms do not imply the given circles have any point of intersection at all.  And you can easily see the have no such intersection in $\mathbb{Q}^2$ since the coordinates would have to be irrational.   
If you want to live with $\mathbb{Q}^2$ as a geometry and just say it does not allow all the usual constructions that Euclid uses, then you have to say one construction it does not allow is finding the intersection points of the circles in Prop 1.1.
The constructible numbers are called constructible because you can get them by compass and straightedge constructions.  The point of adding numbers $\sqrt{1+x^2}$ is to get things like the length of the diagonal of a unit square which of course is $\sqrt{2}=\sqrt{1+1^2}$.  The coordinates of the intersection point in Euclid's Prop 1.1, if you take the base line as going from $\langle -1,0\rangle $ to  $\langle +1,0\rangle$ are constructible as $\langle 0, \sqrt{1+\sqrt{2}^2}$ since we have already seen $\sqrt{2}$ is construcible
A: 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...
A: A geometric model for euclidean geometry should be presented as follows:


*

*Euclidean plane be the open unit disk

*A point of E_plane be a point inside the unit circle and vice versa

*A line of E_plane be part of circle inside the unit circle which passes through a fixed point on the unit circle.


It seems that any student can easily prove that sum of internal angles is equal to $180^\circ$ for triangles.
A.V.Farimani
