Foundations of Geometry

Background

When I was confronted with the prospect of teaching Geometry, I thought back to my own experience working through Euclid "for fun" as an undergraduate. The subject is so simple to formulate yet so dense with complicated instructions, I could not imagine a class based around lectures. I wanted to recreate my own experience with Euclid for the students. This lead me to develop what I called the "Euclid Project".

While I anticipated this project would take up a lot of time, it nearly took us the entire term. Sadly the project did not fully meet the requirements of the class for preservice teachers, so we ended the term with a module comparing Euclidean and spherical geometry. This introduced students to the ideas of coordinate geometry, transformation geometry, and similarity.

Euclid Project

The Euclid project began by reading an introduction to Euclid taken from Chapters 1-4 of Geometry: Euclid and Beyond. This introduced students to the idea behind the axiomatic system, some of the gaps in Euclid's axioms, and how this relates to compass-straightedge. Working in a Class Notebook on OneNote we then carefully "translated" the definitions, postulates, and common notions from Book I into terms that we understood. This often meant using letter names for points, lines, angles, and circles to remove ambiguity from Euclid's prose.

We then slowly worked through Book I following the same routine

Translate four Propositions at a time
Assign one propositions to each student
Put students in pairs to find proofs of their Propositions
Students would present their proof to the class
Students then typed proof and submitted it to Class Notebook.
Students give written comments to each other on their proofs
Students revise their proofs as needed

We only made it to Proposition 31 in Book I. But students had complete ownership of these proofs

Spherical Geometry

My predecessor at Hanover purchased a set of 3 Lénárt Spheres for our department. As there is nothing comparable to Euclid for spherical geometry I adapted the following worksheets from some exercises in Geometry: Euclid and Beyond and Non-Euclidean Adventures on the Lenart Sphere. As the class had largely been constructive up to this point, these worksheets continue this trend.

Constructive Postulates: introduces students to using the spherical compass and straightedge on the Lenart sphere. Also motivates some definitions and axioms of spherical geometry in analogy to Euclidean geometry.
Coordinates: introduces students to Cartesian coordinates for the plane and spherical coordinates for the sphere and how to construct them. Also discusses the appropriate parallel postulate for spherical geometry.
Rigid Motions: has students construct rotations, reflections, and translations in the plane and reflections and rotations on the sphere. Also gives some examples of proofs using rigid motions.
Which Propositions are True in Spherical Geometry: Guides students through Propositions 1-28 of Book I pointing out which need new statements, which need new proofs, which have counterexamples, and what aspects of the Euclidean proofs break.
Dilations: has students construct dilations (for natural numbers) and has them prove the similarity theorems of Euclidean geometry. Guides students through counterexamples to the SSS, SAS, and ASA similarity theorems for Spherical geometry.
Angle-Angle-Angle and Area: Guides students through the proof of the AAA congruence theorem for Spherical Geometry.

Highlights

Isosceles Bisectors

As we worked through Book I, there was some confusion over which of the propositions to construct bisectors, perpendicular bisector, or angle bisector were most useful in a given context. Part of this confusion lies in the fact that the construction are all very similar. One student went so far as to claim that all three are clearly the same when working with an equilateral triangle. This led us to formulate our own theorem! We named it the Isosceles bisector theorem. It claimed the following three lines were the same:

The line from the apex to midpoint of base
The angle bisector at the apex
The perpendicular bisector of the base.

This lead us to discuss cyclic chains of implication. Each student wrote a proof of one implication, with two different proofs given for (3)⇒(1).

Parallel Postulate

One of the main themes of the class was how taste (rather than pure reason) dictate the opinions of mathematicians about Euclid. This is clearest exemplified by the opinions of the Parallel Postulate. We visited Hanover's rare book collection to view the oldest editions of Elements. Amazingly each other offered a different axiom and included Euclid's as a bonus Proposition (usually listed with 29).

As we were "translating" Euclid, I asked the students which axiom they preferred Euclid's or Playfair's. The consensus was that we adopt Playfair's instead. It was not until late in the term that we finally reached Proposition 29 that required the Parallel Postulate. Students then showed how Proposition 29 is indeed the contrapositive of the Parallel Postulate and how either is equivalent to Playfair's Axiom.

Spherical Geometry

Students worked in pairs on their Lenart spheres. They repeated all of the Euclidean constructions that were possible using the Lenart sphere. Here is an image of the construction of equilateral triangles. They also attempted, but failed to make an equilateral triangle where one of the sides had length equal to ⅔π . They quickly learned from experience that the main difference between Euclidean and spherical geometry is that the finite area of the sphere forces lines back towards each other.

One particular interesting case is Proposition 16, where a line is constructed of a certain length. On the sphere this line extends so far that it leaves the region that the endpoint needs to fall in to conclude the proof. This is something that is difficult to imagine, but students quickly stumbled upon it while using the Lenart spheres.

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