Induction in Solid Geometry

Last time, I identified 5 problems I thought were crucial to beginning to understand what the Poincare Conjecture was asking. From there, I started  looking at the question of what is the meaning of simple connected,  which lead me to homotopy equivalence and holes in topological shapes. Working with the Poincare Conjecture, we are exploring topics in topology and as such we need some sort of understanding of the topics covered in topology. The issue I’ve been running into while working on this is how to find questions to explain the ideas, so I hope to further explore this problem.

The Minions of Math have covered topology and geometry in different forms. From descriptions of the 3-space layout of the library of Babel, building models to demonstrate the properties of polyhedron, and proving Euler’s Polyhedral Formula. This last one is a nice simple entry point into algebraic topology, because it’s an simple way to see how objects can be similar when boiled down to their mathematical properties. In the case of the polyhedral formula, we examine how many faces, edges and vertices an object has.

The specific problem set I’m looking at comes from chapter 3 of “Induction and Analogy in Mathematics” by George Polya and deals with using induction in order to come to a proof of Euler’s Polyhedron Formula which is that any convex polyhedron’s surface has the characteristic of F – V + E = 2. F is the number of faces, V is vertices, and E is the number of edges. It takes approximately 24 pages and 41 examples to come to this conclusion, but the whole proof isn’t needed. It’s very thorough and in depth and uses 5 pages to guide the reader to the formula, then proceedes to list numerous examples to test the formula against. In these examples, there are things he touches upon briefly that I feel may be more important for the problem at hand. Such as in section 6, he brings up that a doughnut shaped polyhedron does not work with the formula F – V = E + 2. This is important to the problem at hand, because  it is our first example that shapes with holes are not equivalent to shapes without holes. This is an important property in the discussion of topology, since we can change shapes into other shapes but we can’t add extra holes and we cannot close any holes. But, if I was to use any of these questions, I would have to edit and revise them to demonstrate my points.

What these question don’t explain is what ‘simply connected’ means and how to prove it. These question do not go into higher dimensions, they do not explain Ricci Flow and they do not deal with spheres or circles, instead just focusing on polygons and polyhedrons.We are looking to understand if a simply connected manifold in 3-space is the 3-sphere. Obviously, there is still a lot of work to be done before I can present this. I have been collecting more information on the topic to study. There have been a few books, videos, and lectures have been made to explain the discovery to a general audience. Visual aid is nice, and it’s plentiful but math is a hands on activity at heart and people respond well to having some problems to get their hands on. There are the original papers to read through as well, but there is the problem that it is at a level that is probably over my head. Hopefully, more understanding of the history of the topic will unlock new ideas and new places to look for information. And finally, I am working towards a deadline for the description of this meetup: December 5th. Next time I’ll work on a summary of the material I have.