Felix Breuer's Blog

Using Intuitive Geometry - Lecture Summary

This fall semester, I taught the Math 890 course at San Francisco State University. It was the very first course that I taught all on my own. I am currently funded by the DFG, so I have no teaching obligation (in fact, I had to ask the DFG for permission to devote my time to something other than research). But when I was offered this opportunity, I leapt at the chance and volunteered to teach this course, even though, with two 75 minute classes each week, it was a significant amount of work. I wanted the experience, and, after all the years where I was responsible only for my own research portfolio, I was looking forward to taking on some real responsibility. Also, this was a topic course for graduate students, so that I could teach anything that I wanted. What a wonderful opportunity to experiment!

In this post, I will present the idea behind the course I gave, give an outline of the topics covered, provide all my slides and notes for the lecture as well as all the exercise problems, give a list of topics for the seminar, and review how the course, as a whole, went.

Note: The material linked below is known to contain a number of typos.

The Idea

I always wanted to try a different format of a course. With “format” I do not mean the methodology for teaching - experiments in that department are certainly called for, but I wanted to stick to the classical lecture + seminar format (because I actually enjoyed it as a student). What I wanted to experiment with was the content of my course.

Math courses, the way I experienced them at the FU Berlin, typically build a theory from the ground up, in a very detailed fashion. They emphasize rigor over intuition, and have a classical definition-theorem-proof structure. Courses of this “classic” format are certainly important, especially for teaching mathematical rigor, but here, I wanted to do things differently:

  • The course was to cover a large range of topics, with the aim of showing their interconnections,
  • intuition was to be emphasized over rigor,
  • and it was to make extensive use of visualization.

In short, I wanted to give my students a grand tour of the mathematics that I find interesting. Adding courses of this type to the curriculum is important for a number of reasons. I do not want to go into too much detail on this, so here are just a few bulletpoints, in no particular order.

  • For me personally, grand tours are simply more inspiring than develop-a-theory-from-the ground-up-lectures. I want to take a grand tour of the house, instead of spending my time laying bricks for the foundations!
  • Grand tours have more room to present applications of the covered material.
  • Pure mathematics especially are not measured only in terms the applications they have. The beauty, insight and perspective they have to offer are of direct value to us as human beings. However beauty, insight and perspective are tied primarily to the rough ideas and much less to the technical details. So let’s cover more beautiful ideas and less details!
  • Showing the connections between different fields make all of these fields more relevant to the audience!
  • First, I cannot remember a theorem unless I develop some kind of intuition for it. Second, I cannot prove similar theorems on my own, without having an intuitive idea why it should be true. Third, the intuitive idea is often all I remember off the top of my head after a couple of weeks have passed. So why not focus on teaching the intuition in the first place?
  • For me personally visualization is by far the best way to develop an intuition.


The topics that were covered in the lecture + seminar are show in the big mind map at the top of this page. As you can see, the variety of topics is very large. Each of the big headlines in the mindmap could easily make up an entire course on its own! All of these topics are interrelated. Yet, it is hard to come up with a single title for the course that includes them all. “Discrete Geometry”, certainly, is too generic and not entirely accurate. In the end, I decided to call the course:
Using Intuitive Geometry - Using Geometry Intuitively
I like this title, because it makes clear that I care more about methods and intuition, than about giving a comprehensive account of anything. For example, Game Theory is certainly no subset of Discrete Geometry, but methods from “intuitive geometry” do play a big role: The Minimax Theorem in Game Theory is both equivalent to the LP Duality Theorem and it is implied by Brouwer’s Fixed Point Theorem, which are both staples of intuitive geometry. In hindsight, it occured to me that another fitting title would have been:
Adventures with Hyperplanes
Well, maybe next time. Here are the slides and notes from the tour I took this fall through all of these diverse topics:
  • Visualizing Numbers (Gauss sum formula, lattice points in triangles, staircases, Dedekind sums)
  • Polytopes (H- and V-description, faces, permutahedron)
  • Linear Programming (Fourier-Motzkin eliminiation, Farkas lemma, LP duality, simplex algorithm, combinatorial optimization)
  • Game Theory, Social Choice (minimax theorem, Nash equilibria, spatial voting, McKelvey’s theorem)
  • Topological Combinatorics, Fair Division (Brouwer’s fixpoint theorem, Sperner lemma, ham sandwich splitting)
  • Ehrhart Theory, Lattice Points in Polyhedra (Ehrhart’s theorem, Ehrhart-Macdonald reciprocity, Stanley non-negativity, combinatorial applications)
  • Generating Functions (formal power series as lattice point sets and vice versa, enumeration of lattice points using formal power series)
  • Machine Learning (support vector machines, kernel trick, VC dimension)


To go with these lectures, I made up these exercises, containing 2 to 5 problems each:

Seminar Talks

After 10 weeks of lectures, we had 4 weeks of student talks. Some of the talks were in-depth talks on a particular subject, while others were brief introductions to large subject areas. These are the topics I suggested:

  • The Mordell-Pommersheim Tetrahedron. (Source: Computing the Continuous Discretely. Beck, Robins. 2007. Chapter 8.)
  • Power Indices and Spatial Power Indices (Source: Power and Stability in Politics. Straffin. In: Handbook of Game Theory Volume 2. Chapter 32. 1994.)
  • LP and MIP Modelling Tricks (Source: AIMMS Optimization Modelling. Bisschop. 2011. link)
  • Interior Point Methods (Source: The integer-point revolution in optimization: History, recent developments, and lasting consequences. Wright. 2004. link)
  • The Borsuk-Ulam Theorem and Ham Sandwich Splitting (Source: Using the Borsuk-Ulam Theorem. Matousek. 2003. Chapters 2 and 3.)
  • Integer Programming Methods (Source: Theory of Linear and Integer Programming. Schrijver. 1986. Chapters 23 and 24.)
  • Order Polynomials (Source: Combinatorial Reciprocity Theorems. Beck. 2011. Chapter 6. link)
  • P-Partitions (Source: Combinatorial Reciprocity Theorems. Beck. 2011. Chapter 7. link)

How did it go?

I am very happy with how the course turned out! This “grand tour” style of a course succeeded in engaging the students. From what I heard, my students not only learned something, but also took away a new perspective on many of the subjects I covered. One student even told me that they now had a much better understanding what the thesis they had been working on is about. Was will man mehr!

Nonetheless there are of course a number of things I would like to improve the next time I teach this course - which I would really like to do! Here is a list of notes that I would like to keep in mind for the next iteration.


The exercises were what made this “intuitive tour” type of course work. Had my students not spent the time to accquaint themselves personally with the subject matter, I guess many parts of this course might have remained too vague. The great thing is, that my students did work through virtually all of the exercise problems, even though I did not have the resources to grade the correctness of their solutions. It would have been too easy to skip corners - but they didn’t! Thus, I really have to thank them for their initiative, curiosity and dilligence - they made this course work (even more than ususal)!

In hindsight, I think this large program is better suited to a course with three 75 minute sessions per week. Two sessions for lectures and one for talking about the exercise problems. Also, it would have been very valuable to provide individual corrections for the exercises and/or solutions of the exercises for the students to review on their own.

Lecture Notes

I wrote the script for my lectures as I went along. Given that the material was so diverse and scattered over so many sources and given that it was my ambition to present some material from an (AFAIK) original perspective, this was very work intensive. Originally, I had planned to use my blog to record detailed lecture notes, but this turned out to be infeasible. (Currently, it takes me at least four hours to write up notes for one 75-minute lecture in a form suitably self-contained for a blog post - even if I have prepared the material beforehand in slide form and given the lecture in front of class. Is that typical or somewhat slow?)

The points mentioned so far (publish lecture notes, publish solutions for the exercises, provide individual corrections for the exercises) all boil down to putting in more work. And when I give this course the next time, I certainly want to prepare lecture notes and solutions beforehand and work together with a teaching assistant for grading! Here are some more thoughts, though, that take a different direction:


Combining the lecture with a seminar was an excellent choice. For two reasons. First, it gave the students the opportunity to work on a subject in depth. This was especially important as the “intuitive tour” format (by necessity) left the students wanting more details. The seminar gives students the opportunity to pick one subject they find particularly appealing and dig into the details that were only hinted at in the lecture and in the exercises.

Second, the seminar was enjoyable not only for the students, but for me as well! During preparation, it was simply fun to get together and puzzle about a problem or figure out the best picture for illustrating a theorem. (Especially after I had spent hours on end with mainly one-way communication during the lecture.) On the whole, the talks were very good and got across all the important points. Some even went above and beyond the suggested material! Next time, I might be tempted to make the seminar a much bigger part of the course!


What place do proofs have in this type of course? After the first two weeks of the course I become worried that my intuitive style of lecturing might be too vague for my audience. This worry was not fed by any actual observations during class, but rather by my rookie-lecturer-confidence wavering. So I decided to up the level of detail in the next few weeks and include more proofs and a greater level of formality. In retrospect, after talking with my students about it and judging from the their exercise solutions, I can say that this failed completely. This “proofy” part of the lecture was the part my students enjoyed the least and it was the part they had the least grasp of, judging by the exercises. (Of course, proper lecture notes would have helped their performance in the exercises, but I think this is not the main point here.)

I was reminded here of these great quotes by Victor Klee:

  • “A good talk contains no proofs; a great talk contains no theorems.”
  • “Mathematical proofs should only be communicated in private and to consenting adults.”

In any case, they have no place in this type of lecture! Intuitions for why a theorem should hold, rough proof sketches, all of these certainly belong in this type of lecture. But going the extra distance from an intutive proof sketch to an actual “formal” proof is not worth it - simply because any formalization is significantly harder to understand than the intuitive idea that gave rise to it. If your goal is to teach a theory, then teaching that formalization is part of what you want to communicate. If your goal is to show connections between different fields intuitively, then that formalization is only tangential to what you want to communicate. So, adding formalism just for the sake of communicating an intuitive idea is not going help - instead of clarifying the point it will obfuscate.

Of course this does not mean that proofs are irrelevant! On the contrary, if students are to learn how to use geometric intuition, then learning how to make intuitive ideas precise is crucial. However this is an active process. Students have to do it themselves when solving exercises or preparing seminar talks. My doing it at the whiteboard does not help anybody! More precisely: my doing it at the whiteboard helps only, if students have tried to do it by themselves beforehand.

I will not try to write down here all the mental notes I made for myself on how to present proof ideas if you do not want to make them 100% precise. (Maybe I will write another blog post about this.) I want to mention two things, however: First, when explaining an algorithm/constructive proof it is much more important to make the construction precise and illustrate it with an example than to prove its correctness. Second, always give a concise summary of the key point of a theorem or proof that you just explained. That way, even those members of your audience that you may have lost will have something concrete to remember.


Of course, when talking about geometric intuition, drawing is the most important thing! Two shorts thoughts about this:

The most important aspect of many mathematical figures is the way they are drawn and not the final image. Thus it is very important to construct a figure slowly, step by step. No fancy visualization tool beats a good drawing on a whiteboard!

In this vein, I encouraged my students to draw the objects they were talking about, both in the exercises and in the seminar - and I got everybody to do it in the end! I was very happy about this, because, in my opinion, drawing is an essential skill for any mathematician!


It was a great experience teaching this course! I learned a lot, both in terms of hard skills and soft skills. I would love to teach this course a second time, and the next time I would like to experiment more with teaching methodology. Also, I would really like to write up this tour through some uses of intuitive geometry in book form and smooth out all the wrinkles along the way. But, maybe, the printed page (or static PDF document) is just not the right medium for this.

I think mathematics could benefit from more courses and texts of this “intuitive tour” type: overview material with a strong emphasis on intuition and visuals, accessible to advanced undergraduates but with enough depth to satisfy graduate students.


Finally, here is a short informal summary of the literature I used for the preparation of this course.
  • Beck, Combinatorial Reciprocity Theorems.
  • Beck, Robins, Computing the Continuous Discretely.
  • Bisschop, AIMMS Optimization Modelling.
  • Breuer, Heymann, Staircases in , Integers.
  • de Longueville, A Course in Topological Combinatorics.
  • Gaiha, Gupta, Adjacent Vertices on a Permutohedron, SIAM Journal on Applied Mathematics.
  • Grötschel, Lineare und Ganzzahlige Programmierung.
  • Matousek, Using the Borsuk-Ulam Theorem.
  • McKelvey, Intransitivities in multidimensional voting models and some implications for agenda control, Journal of Economic Theory.
  • McKelvey, General Conditions for Global Intransitivities in Formal Voting Models, Econometrica.
  • Raghavan, Zero-sum two-person games, Handbook of Game Theory, Vol 2.
  • Schölkopf, Smola, Learning with Kernels.
  • Schrijver, Theory of Linear and Integer Programming.
  • Straffin, Power and stability in politics, Handbook of Game Theory, Vol 2.
  • Wright, The interior-point revolution in optimization: History, recent developments, and lasting consequences, Bulletin of the American Mathematical Society.
  • Ziegler, Lectures on Polytopes.