12 pentagons!

I’ve been reading Sternberg’s “Group Theory and Physics”, Cambridge University, reprinted 1999. On pages 43 to 44 he says, “… every fullerene has exactly 12 pentagons. This is not an accident.”

The stable structure of carbon which has 60 carbon atoms arranged like the vertices of a soccer ball is called a buckyball. It turns out that, in similar structures, we can have any even number, greater than 18, of carbon atoms except for 22. This is equivalent to polyhedra having 12 pentagons and any number of hexagons except 1.

This family of structures consists of polyhedra whose faces are either pentagons or hexagons. In chemistry they are labeled by the number of carbon atoms, so they talk about

I find it unforgettable and marvelous that the number of pentagons is always exactly 12. And I can prove it.

Simplicial curvature & simplicial Gauss-Bonnet

Introduction and text

The last section of Bloch’s chapter 3 (simplicial surfaces) is a long (and to my mind at this time, uninteresting) proof of the 2D Brouwer fixed point theorem: any continuous map from the disk to the disk has a fixed point. Bloch also proves a corollary, the no-retraction theorem, that there is no continuous map r from the disk to the circle such that r(x) = x for all x on the circle.

That one sounds interesting. We’ve seen in before, with the commentary that you can’t map the surface of a drum onto its rim without tearing it. I still don’t see it that way. But it is rather shocking that the map r cannot preserve all the points on the rim.

Anyway, we’re not going to fight with those. For me, the climax of chapter 3 is the simplicial Gauss-Bonnet theorem. It shows that there is a definition of curvature for simplicial surfaces (in fact, for polyhedra in general) such that the total curvature of a surface is equal to times its Euler characteristic .

(A simplicial surface is a polyhedron all of whose faces are triangles. I expect we’ll see this again in another post.)

That the total Gaussian curvature of a surface is equal to is called the Gauss-Bonnet theorem. It is a reasonable culmination of a first course in differential geometry. The simplicial version means that we have a definition of curvature for simplicial surfaces and polyhedra which gives us a form of the Gauss-Bonnet theorem. That says it’s a reasonable definition of curvature.

So what is this marvelous definition of simplicial curvature? It’s also called the angle defect, and goes back to Descartes.

The Euler Characteristic: Teasers

These are things i came across when I first started looking at the Euler characteristic, in fact, when I was looking at triangulations in particular.

n-manifolds

The Euler characteristic generalizes to dimensions other than 2, and there are at least three noteworthy theorems involving the Euler characteristic. I’m not going to say much about them, because they, like so much else, are still outside my comfort zone. I’ll just barely tell you what they are, and leave you to chase them down if they interest you.

As we’ve seen, the Euler characteristic of a polyhedron is given by

,

where v, e, f are the numbers of vertices, edges, and faces. Homeomorphic polyhedra have the same Euler characteristic, and that means we can define the Euler characteristic of a topological surface as the Euler characteristic of any polyhedron which is homeomorphic to it.

This alternating-sign sum of the numbers of 0-, 1-, and 2- simplices generalizes in the obvious way: for an n-simplex, we take the sum, with alternating signs, of the numbers of k-simplices, for k <= n. As for surfaces, so for n-manifolds: this is a topological invariant, and we want to define the Euler characteristic of an n-manifold as the Euler characteristic of any k-simplex homeomorphic to it.

surfaces: visualizing the gluing of them

Quite some time ago, a friend asked me what would happen if we tried to construct a torus by gluing all 4 sides of a sheet of paper together, instead of first one pair then the other. Didn’t the math have to specify first one pair then the other?

One reason I’ve been hesitating over this post is that it doesn’t seem to be “real” mathematics – though any number of people might howl that PCA / FA isn’t “real” mathematics either. This is just a small drawing that I cobbled together to show that the homeomorphism between a circle and a line segment with endpoints identified… well, it doesn’t have to correspond to a physical process. (Why don’t I refer to “the glued line”?)

“… algebra provides rigor while geometry provides intuition.”
from the preface to “A Singular Introduction to Commutative Algebra” by Greuel & Pfister

It helped me to go back and read my original comment when I acklowledged Jim’s question to this post. I see that I did not understand that what matters to the formalism, the algebra, is before and after; what matters to the geometry is between or during. We reconcile them by permitting some things in the geometric visualization that we would not permit in the formal algebra, if the algebra even formalized the process: points passing thru points; or even some tearing, if when we reglue it we restore it rather than take the opportunity to change it.

That’s what bloch says, on p.57, discussing the physical process of getting from the knotted torus

to the regular torus.

The Euler characteristic (triangulations 2)

I want to talk about some things I saw when I was looking at triangulations. This also continues my reading in Bloch.

The major thing I saw was the Euler characteristic. We usually define it for polyhedra, as the alternating sum / difference of the number of vertices, edges, and faces (0-, 1- and 2- simplices)…

Then we would define it for a surface by taking the Euler characteristic of any polyhedron which is homeomorphic to that surface.

For that to be well-defined, of course, requires that all polyhedra which are homeomorphic to a surface S have the same Euler characteristic (as they do).

By the same token, we could define the Euler characteristic of a surface from any triangulation of the surface, after we show that all triangulations of a surface have the same Euler characteristic. Oh, and we’d better actually prove that every topological surface (every topological 2-manifold) can be triangulated.

Yes, they can be. Not true for topological 4-manifolds, and I think it’s still wide open for higher dimensions. In contrast, I believe that every differentiable n-manifold supports a unique piecewise linear (PL) structure (which is the generalization, they tell me, of triangulations).

Details.

Triangulations of Surfaces: minimum number of triangles

Edited 4 Sep. search on “edit”.

Take a cube. If you cut it along a few edges, you could lay it out flat. To restore the cube, we identify certain edges with each other. Similarly for a tetrehedron, or a theoretical soccer ball (with flat faces), or any polyhedron. For studying surfaces by looking at polyhedra (i.e. picewise linear structure), it is convenient to use only triangular faces (2D simplices) rather than arbitrary polygonal faces. The analog of our cut and flattened cube is called a triangulation. As before, we want to identify certain edges with each other.

In particular, the following

is offered as a triangulation of a torus (with the top & bottom edges identified, and the left & right, as we’ve seen before).

Why are there so many triangles? I have wondered that since the first time I saw it.

Topological Surfaces: Bloch Ch 2.

Let me talk about the second chapter, “Topological Surfaces” in Bloch’s “A First Course in Geometric Topology and Differential Geometry”. I finished it quite a while ago, but I’ve had trouble deciding how to talk about it. I don’t want to just summarize it. Instead, I think I’ll try asking and answering some leading questions.

First of all, why did I choose to read Bloch? A few years ago, I was seriously shaken up by the following, from Freed & Uhlenbeck’s “Instantons and Four-Manifolds”, p. 1: “A basic problem is to ascertain when a topological manifold admits a PL [piecewise linear] structure and, if it does, whether there is also a compatible smooth [differential] structure. By the early 1950’s it was known that every topological manifold of dimension less than or equal to three admits a unique smooth structure.” They were setting us up for the fact that it isn’t generally true in dimensions greater than 3.

To be explicit about the challenge we face: topological, simplicial (which Bloch tells me generalizes to PL), and differential structures on a manifold coincide for low dimensions but not for high, and “high” means “greater than 3”.

This was all news to me. I knew, for a small value of knew, differential geometry, and I’d seen some simplicial stuff – triangulating and cutting up surfaces – and I was acquainted with topology, possibly for a large value of acquainted, but their “basic problem” was nothing I’d ever heard before.