Greetings fellow math majors and professors of Canisius College. For three years now I have been conducting independent research in the the field of Euclidean Geometry. Specifically, Applications of Euclidean Postulate 9: The Plane Separation Postulate. The purpose of my research was to develop a theorem that tells the number of disjoint subsets there are in a given set, when lines are drawn to connect two different boundary points. After many unsuccessful attempts at creating such a theorem, I do believe that I have finally developed such a theorem. Based on the number of straight lines and lines involved in each cross, the number of disjoint subsets can be calculated. Although the foundation of this theorem is geometrical, I will also show how this theorem can help solve simple algebra problems. I will also be presenting a blueprint on how this theorem may be adapted to curves if further research is to be conducted.

**Theorem:** Let Q be a Jordan curve along with its interior. The number of disjoint subsets inside the Jordan curve is represented by **S=(n+1)+sum(gamma_p_i -1)** where Q is cut by n lines which can intersect at a finite number of points $p_1, p_2,…,p_K so that gamma_p_i denotes the number of lines meeting at point p_i.

Hope you guys enjoyed my presentation and learned quite a bit about euclidean geometry, topology and possibly what it takes to one day create your own theorem.

Wow! Your presentation really blew me away! Who would have ever thought that a mathematical concept so complex would originate from thinking about how to divide a pizza? I really liked all of the drawings that you did on the board when you gave your presentation. They really helped to clarify and visualize exactly what was going on. I know that you said that working on developing a formula and figuring out the proofs was a painful process, but all of your work really showed in this outstanding presentation! Also, it was really great that you actually got to present it to us considering how hard you worked! Since I’m new to this, your presentation really helped me to understand what seminars were all about, so thank you!

Great presentation Jake! I remember when you first told me about this. It has come a long way since then. It’s amazing how this idea formed because you were thinking about pizza. It was a great presentation and really well illustrated. I’m glad there is more to do. It’s great to have an unfinished problem so you can see how others will solve it. Now ideas could go back and forth on how to finish the rest of your research. Great idea!

Great job, Jack on your presentation. I enjoy your sides when you were presenting. I like the confidence that you had when you were displaying your ideas to us. I learn something new from you are presentation because you show it to us that having a simple-minded concept can lead to great achievement in the future. Whoever thought that your simple idea of pizza can lead to great research or finding on the Application of Planar Separation. I am still learning, and I hope to learn more from you. Thank You.

Sorry for Name spelling Jake. I mistype you are named.

It is truly amazing to think that math can be found in the simplest things! It is admirable to see the dedication and persistence you demonstrate by pursuing your own research. Your presentation was very informative and your graphics in the presentation and on the board truly helped me follow the logic and reasoning. In hindsight, Math Club totally should have bought you a pizza for a live demonstration of your theorem! In all seriousness, I was curious as to the applications of your theorem to cartography and drawing property lines. It would be interesting to see if another application of your theorem could be used to divide a plot of land owned by x amount of owners.

Due to the formula not being 1 to 1 it’ll be very hard to show or use that as an application. If there are x land owners then there would be a maximum x-1 lines and minimum number of lines you’d get would be (x^2+x+2)/2 which is the Lazy Caterer’s sequence. So you the amount of different ways you could arrange the different cuts of land would be ((-x^2-x)/2)! for each x.

Correction: While the maximum amount of lines you would need to create x spaces would be the Lazy Caterer’s sequence and the minimum amount of lines you would need to create x slices would be (x-1) the amount of different configurations for x slices would just be x^2-x.

But if you had to do a problem like which you described it would depend on the areas of the geometry of the area to formulate the best way to get equal areas.

Great presentation to watch and I appreciate the organization of it all with supporting theorems and lemmas alongside examples and several types of proofs. I’d be curious how to tackle this same problem with non-Jordan curves. What if I get a doughnut-shaped pizza? Or better yet a Möbius strip pizza?! I’d love to see it extrapolated to higher dimensions as well.

Regarding the doughnut the idea, I honestly don’t know at all because if a line went from outer curve to outer curve it’d follow the theorem. Where it goes sideways is where a line goes from outer curve to inner curve. The mobius strip would be a continuation on the doughnut problem, but since a mobius strip is in three dimensions the lines would have to be planes because the cutting agent must be of the same dimension as the thing being cut. I proved this in one dimension in my report.

Being the first presenter, you really set the bar high for the rest of us. I really liked that you started out with the pizza problem. As your presentation got denser and more complicated, the pizza reference made it easier to wrap my mind around what you were saying. You mentioned how long you have spent working on this talk, and it really shows.

Great presentation Jake! I remember you telling me about it last year and proud of how far you’ve come with it. You explained everything well and made it easy to understand. Who would have thought that thinking about pizza would lead to this. Can’t wait to see if you can ever build off of this in the future.

With so many others, I remember the beginning of this idea as well. It must be really rewarding to see it through to this point! I think the presentation set a high bar but also demonstrated how a simple pizza problem can turn into a complex math journey. But applying yourself always pays off, and this was a very impressive presentation. Great job!

An absolutely fantastic presentation, and I really enjoyed seeing the complete work, especially after seeing all the work you put into it in the Math Lounge. Super cool way of showing how math is really a fundamental concept of basically everything. Congratulations!

I really enjoyed your presentation. I am new to this but I noticed that math could be found from something simple. I was motivated after knowing hardness from your passion to create or develop a theorem. Great job!