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This question is a follow-up to that one, where I expressed doubt about the use of abstract affine geometry in undergraduate education.

However, future high-school teachers need to be able to relate their higher education to what they will teach, and they will have to teach affine geometry without much formalism available (e.g. they should also be able to explain how vectors relate to points without resorting to $\mathbb{R}^n$ in the first place).

A teacher should be able to use the linear part of affine transformations to quickly realize that a composition of a rotation of angle $\theta$ and a rotation of angle $-\theta$ with different center is a translation, even if he or she does not expect his or her pupils to come with that conclusion: this is an example of why teachers should know more than what they teach, so I assume an undergrad curriculum with quite some geometry.

My question is the following:

What is a realistic and convenient framework for affine (possibly Euclidean) geometry which can be used by a high-school teacher, but relates easily to undergrad-level math curricula?

A possible answer could be a set basic objects (points, lines, etc.) and of axioms to work with, but maybe there are better, non-axiomatic ways of achieving this goal. It seems difficult to have enough axioms to cover all what one would have to teach in high-school, while not so many that it would be intractable.

I seek answers that would be addressed to the teacher-to-be, not to its pupils; however it should be relatively easy to use the given framework in high-school teaching. I would already be happy with propositions that only cover plane geometry.

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    $\begingroup$ Affine geometry with rotations is metric geometry, since we can define "$P$ is equidistant from $Q$ and $R$" by "there is rotation around $P$ that takes $Q$ to $R$". $\endgroup$ – user173 Apr 23 '15 at 12:41
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On the axiomatic front, you might find the axiomatization of Desarguesian planes useful for your project. A very nice exposition is chapter 2 of Emil Artin's "Geometric Algebra", which I'll now summarize.

The undefined notions are points, lines, and an incidence relation between them. The axioms for an affine plane are

  1. Through any two points there is a unique line.
  2. Through any point not on a line there is a unique parallel line through that point.
  3. There exist three points not all on the same line.

Using these three axioms (the second really) one can show that being parallel is an equivalence relation on lines. One can then define an affine transformation $T$ to be one that preserves this relation, i.e. if $\ell_1\parallel \ell_2$, then $T\ell_1\parallel T\ell_2$.

More significant are the affine transformations that leave the parallelism class invariant, i.e. such that $\ell\parallel T\ell$. Artin calls such affine transformations dilatations. This is because dilatations are uniquely determined by where they send two points, as then the location of the third point is uniquely determined by the parallel postulate.

Consequently, he defines a translation to be a dilatation that has no fixed points, and a dilation to be a dilatation that has one fixed point. One can show that dilations with no fixed points are uniquely determined by where the send a single point. This allows us to visualize the action of translations on the plane as arrows: an arrow for a translation indicates that the translation takes the starting point to the ending point. It is instructive to prove that any two non-collinear arrows that come from a translation can be realized as opposite sides of a parallelogram. This is the affine version of the statement that translations are specified by the data of a magnitude and direction.

We do not yet know that any arrow comes from a translation, however. This is precisely the content of axiom 4a of Desarguesian' geometry.

4a. Desargues' theorem for parallel lines. Given three distinct parallel lines $\ell_1\parallel\ell_2\parallel\ell_3$, suppose we have points $A_i,B_i$ on $\ell_i$ so that $A_1A_2\parallel B_1B_2$, $A_2A_3\parallel B_2B_3$. Then $A_1A_3\parallel B_1B_3$.

The point is that if the arrow $\vec{A_2B_2}$ were to come from a translation, then that translation would have to send $A_1$ to $B_1$ and $A_3$ to $B_3$. But this is actually a dilatation if and only if $A_1A_3\parallel B_1B_3$, which is exactly what this version of Desargues' theorem asserts.

4b. Desargues' theorem for concurrent lines. Given three distinct concurrent lines $\ell_1,\ell_2,\ell_3$ with common point $O$, suppose we have points $A_i,B_i$ on $\ell_i$ so that $A_1A_2\parallel B_1B_2$, $A_2A_3\parallel B_2B_3$. Then $A_1A_3\parallel B_1B_3$.

This second version of Desargues' theorem says that giving the points $O,A_1,A_2$ does in fact specify a dilation fixing $O$ and sending $A_1$ to $A_2$.

An affine plane satisfying axioms 4a and 4b is called a Desarguesian plane. These are significant because the translations plus the identity dilatation form an abelian group, and conjugating these by a dilation results is an endomorphism of this abelian group that preserves the trace lines of the translations. As Artin describes in detail, together with choosing the three non-collinear points $O$, $A$, $B$ from Axiom 3, this establishes a correspondence between Desarguesian planes and $2$-dimensional modules over division rings. In other words, the 5 axioms above are all you need in order to relate synthetic geometry to coordinate geometry.


Artin does further cover the fact that requiring an order relation on the points on the line of a geometry corresponds to requiring that we have an ordered division ring, which ends up being an ordered field, so something inbetween $\mathbb Q$ and $\mathbb R$. What Artin doesn't do is discuss lengths, congruences, and similarities. I worked out some of this some time ago, but I'm not full satisfied with it. In any case, the relevant axioms for lengths and angles are

A. A translation-invariant distance function satisfying the triangle inequality (i.e. a norm on the vector space of translations).

B. The axiom that across any line there is a distance-preserving reflection, where a reflection is an involutive affine map fixing that line. This establishes the necessary relation of perpendicularity (equivalent to a positive definite inner product on the vector space of translations).

This may seem strange, but remember that the affine plane is still the affine plane even if you look at it at an angle. Then what used to be circles now look like ellipses, but all of geometry still holds. You can get rotations as composed reflections, and this is enough to recover traditional side-angle-side congruence and angle-angle-angle similarities.

P.S. Vectors are translations.

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    $\begingroup$ That is interesting, but I do not see how it could be taught to the kind of student I have. It seems to fall on the "too few axioms" side of the delicate equilibrium I am seeking, making the theory two difficult to motivate and relate to both ends we are to tie together. $\endgroup$ – Benoît Kloeckner Apr 22 '15 at 8:24
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I would axiomatize affine geometry using the single function "go $x$% of the way from $P$ to $Q$", where $x$ is a real number.

You can get a feel for this approach in a paper by Suppes available online. He also refers to a book which may be your best source: Szmielew's "From Affine to Metric Geometry".

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    $\begingroup$ Other references which take this approach to affine geometry include Applied differential geometry by Burke, and A vector space approach to geometry by Hausner. $\endgroup$ – Mike Shulman Oct 12 '15 at 16:02
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There is a long list of definitions of "affine space" here; I don't know whether any of them fits your needs.

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  • $\begingroup$ Probably not, but nice list. $\endgroup$ – Benoît Kloeckner Oct 12 '15 at 19:42
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    $\begingroup$ Those are all the definitions of affine space that I know. If you find another one, maybe you could add it to the list! $\endgroup$ – Mike Shulman Oct 12 '15 at 23:14
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In "that one" question you reference, you mention becoming an examples based mathematician and teacher. Examples and how they are used are part of what needs to be taught, and (I believe) are more easily grasped by the bulk of the classroom than (a presentation that heavily favors) an axiomatic framework and derivation. The framework you seek is a theory which allows a form of presentation of the examples so that the students can derive them as well as learn them. (The one thing I remember from a projective geometry class that I graded was an application for railroad tunnel making: how to extend a line past a bounded obstacle using geometric construction.)

The teachers need to know a list of important examples and the accompanying derivations. You can use an axiomatic framework as a basis, but for the teacher to internalize and communicate it, you need (could really use to great effect) mottos and mnemonics. For the translation by two rotations, you can use "Ballerina twirls across the room" to convey the idea. You can devise similar mnemonics to communicate key ideas of the result derivation from the axiom system you care to use.

Once you have a list of such results, mottos, mnemonics, and derivations, you can prioritize and see which ones give you the greatest pedagogical mileage. Present those to the teachers, and assign the rest as exercises.

Gerhard "Mentally Dances Through The Answer" Paseman, 2015.10.12

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