What can we measure? Part I

I recently mentioned this course on Discrete Differential Geometry. One section, whose title I have shamelessly stolen, discusses a curious result known as Hadwiger's theorem. What it doesn't include, however, are explanatory pictures. So read on for a an elementary pictorial introduction to Hadwiger's theorem.

Consider finite unions of convex bodies in Rⁿ. Write the volume of such a set, A, as v(A). Here are some properties of v:

1. v maps finite unions of convex bodies to real numbers.
   
2. v(A ∪ B) = v(A)+v(B) is A and B are disjoint.
   
3. v is invariant under rotations and translations.
   

These are fairly general properties so an obvious question is this: can any other function, besides volume, satisfy these properties? Hadwiger's theorem says "yes" and shows that in Rⁿ there are n+1 distinct functions, v_r for r = 0,…,n and all others are linear combinations of these generalised volumes. v_r corresponds to the usual volume and in general we say that v_r computes the r-volume. So what are the r-volumes for dimensions less than n?

According to Hadwiger, if R is a closed cuboid of dimensions x₁×x₂…×x_n then its r-volume is the sum of all possible distinct products of r of the x_i. For example if n=4 then the 2-volume of the cuboid is given by x₁x₂+x₁x₃+x₁x₄+…+x₃x₄. The r-volume is simply the product x₁x₂…x_n and the 0-volume is simply 1. But...if you're thinking like I was when I first looked at the article you'll be thinking that this can't make any sense - how can the number 1 represent any kind of generalised volume because it's a constant and hence can't be additive?

Hadwiger's theorem is hard to prove. But for now I'll specialise to 2 dimensions and do just enough computation to show that the generalised volumes do make some kind of sense. And I'll show that the 0-volume turns out to be a variation on a well known topological invariant that you may have seen before.

One more bit of preparation. Let me use =_r to mean equality of r-volumes.

So here's a computation of the 2-volume of a rectangle, R, using v₂(R) = x₁x₂:


Here's the 2-volume of a square:


And it's straightforward to see that additivity holds in this case:


So now consider the 1-volume of a square:


And the 1-volume of a rectangle:


Unfortunately, Hadwiger's theorem appears to be blatantly false if we cut a rectangle in half:


So what's going on? The important thing is that when we cut a rectangle in half we have to choose where the points along the cut go. (For example consider cutting the interval [0,1] in half. We get either [0,0.5)+[0.5,1] or [0,0.5]+(0.5,1].) The way I drew it above the points along this line were getting attached to both of the pieces. So the rectangle wasn't the disjoint union of the two squares I drew. So I'll use the convention that a dotted line around a shape means that the edge itself isn't attached. So we get:

Which allows us to deduce:

and hence from additivity we get

and then

and (with apologies for my scanner's cropping):

Unlike with the usual 2-volume, infinitesimally thin edges do contribute. But this should lead you to worry about another issue: in the above working I handled edges carefully, but not individual points. Rather than go back and work carefully with edges I'll just say that they don't contribute to 1-volume but they do contribute to 0-volume. So I'll now move onto 0-volume with correct handling of points and I'll leave it as an exercise to rework the above showing that it is actually correct.

I'm using a small dot to mean the point is included and a small circle to mean a point is excluded. (I hope you can tell a small circle from a zero!) The 0-volume of a closed rectangle is 1. Sounds too trivial to be interesting? Well, from here onwards I think just the pictures will do:










And here's the result I've been aiming for:


In other words, the 0-volume of a closed shape is #vertices-#edges+#faces. Sound familiar? It's nothing other than the Euler characterstic. If it's not familiar, there are countless great articles on the subject out there.

Anyway, to be continued...

Update: Baez talks about some of this stuff here where it's related to Schanuel's characterisation of the Euler characteristic. Yet again I can't write about anything without JB haven't beaten me to it. I could swear that guy's following me around... :-)