Blogging from The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics

Dr Alice Hui from The University of KwaZulu-Natal.

Projective geometry as an undergraduate course: A tour of the three worlds of mathematics.

Attempt to persuade the audience to have projective geometry as an undegraduate course.

**What is Projective Geometry?**

Projective geometry describes objects as they appear, rather than as they are.

Two sides of a train track are parallel but look interest in the far end.

Taken from https://plus.maths.org/content/sites/plus.maths.org/files/features/projective/tracks1_small.jpg

- Advantages of having projective geometry as an undergraduate course
- An axiomatic approach to study geometry
- A projective plane satisfies three axioms.
- Can allow students to study geometry using an axiomatic approach.
- Comparison to other axiomatic formulations that may be taught in undergraduate courses:
- We use axioms to study natural numbers, real numbers, group, ring, field, vector space, set theory. They are not geometric.
- Euclid’s Element is not rigorous in modern sense.
- Topology is a good course but a bit too advanced. Projective geometry is an alternative course to axiomatically study geometry.

- Projective geometry gives a new perspective on Euclidean geometry. To turn the ordinary Euclidean plane into a projective plane, proceed as follows:
- To each class of parallel lines add a single new point. That point is considered incident with each line of the class. Different parallel classes get different points. These points are called the points at infinity.
- Add a new line which is considered incident with all the points at infinite (and only them). This is the line at infinity

- An axiomatic approach to study geometry

**Example 1**

Desargues’ theorem. Two triangles are perspective with respect to a point, if and only if, the intersections of corresponding triangle sides are colinear.

See Burkard Polster and Marty Ross – how to make a perfect plane

In the Euclidean plane, for any triangle, the orthocenter, circumcentrer and centroid are colinear. Can use Desargues’ theorem to prove this. (Jürgen Richter-Gebert, Perspectives on project geometry: A guided tour through real and complex geometry)

**Example 2**

Conic sections are indistinguishable from one another in projective geometry:

A theorem on the real projective plane can be applied to an ellipse, a hyperbola, a parabola, a pair of intersecting lines, and parallel lines in the Euclidean plane.

Pascal’s Theorem. If a hexagon is inscribed in a conic, the three pairs of opposite sides meet in collinear points.

Different illustrations of the Pascal’s Theorem to the conics in real plane:

Example 3

Studying projective geometry at undergraduate level is a good preparation for teaching in high school:

“College algebra, trigonometry, analytic geometry and calculus are a better preparation for teaching algebra than they are for teaching geometry. They involve almost constant algebraic manipulation even when their subject matter is not algebra.”

“The geometry of high school is synthetic”

“Synthetic projective geometry is more closely related to the geometry of Euclid than any other branch of mathematics. It gets the student back to synthetic methods. It involves more of reasoning and less of the technique of manipulation than either analytic geometry or the calculus. ” **Reference: W. H. Bussey, Synthetic Projective Geometry as an Undergraduate Study, American Mathematical Monthly, Vol. 20, No. 9. (1913), pp. 272-278**

**Three mathematical worlds**

- Embodied world (Based on human perceptions and actions in a real-world context)

Use dynamic software

See the GeoGebra book: https://www.geogebra.org/b/113535#

2. Perceptual world: start with analytic definitions of projective planes

3. Formal world: (Starting from axioms, make logical deductions to prove theorems)

Work with axiomatic definition of projective planes

Reference: David Tall (2004). Thinking Through Three Worlds of Mathematics. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281–288.

Suggested textbooks

1) R. Casse, Projective Geometry: An Introduction, Oxford University Press; 1st edition (2006).

2) H.S.M. Coxeter, Projective Geometry, Springer, 2nd edition (2003). (With a GeoGebraBook)

3) C. R. Wylie Jr., Introduction to Projective Geometry, Dover (2008).

**Suggested topics for student projects**

- Partition of finite projective spaces by lines: Start with Kirkman’s schoolgirl problem.
- 15 schoolgirls who always take their daily walks in rows of three. How can it be arranged so that each schoolgirl walks in the same row with every other schoolgirl exactly once a week?
- Two of seven solutions to the problem are partitions of the finite projective space PG(3,2) by lines
- See http://demonstrations.wolfram.com/15PointProjectiveSpace/:

- Non-existence of projective planes of small order:
- Based on a long-standing conjecture: Any finite projective plane has a prime power order. (The smallest open case is 12.)

Joan RaynerNovember 28, 2015 at 10:48 pmThe book looks good. There is very little out there that gives a non-algebraic picture.

I teach conic sections to grade 9 before they come to them in algebra, giving them an understanding of these curves as different “versions” of the same thing. Then I teach projective geometry at a grade 11 level, so though it is not rigorous in proofs, the students have a good sense of paralellism and duality and we cover most of what is in this book. Thank you for sharing.

Jonathan ShockNovember 29, 2015 at 5:01 pmHi Joan, this sounds like a really innovative curriculum! How much freedom do you have to teach what you want?

Joan RaynerNovember 30, 2015 at 7:30 pmThis is part of the Waldorf curriculum. Steiner indicated that projective geometry is appropriate to encourage creative thinking – the thinking needed for the future of humanity. He studied it at some level, but did not develop a high school curriculum. That has been done subsequently. There are some text books, mostly in German, but, as with all the teaching in a Waldorf school, I have an enormous amount of freedom to bring what I think may be appropriate to the students of a particular class. I work basically with other Waldorf teachers both here and in Germany.

Joan RaynerNovember 28, 2015 at 10:50 pmPS – I think projective geometry should be taught in schools as a start, as well as an undergraduate course.