Upcoming Films

Ratatouille (A French rat chef who can cook complements a French human chef who has hands and is allowed in restaurants)

I generally like animated films. And I used to like stories about talking rats. But somehow I felt I'd reached a limit, and this looked insipid. However, several people watching it had said it was very good, so maybe I'll have to see it after all.

Transformers

I quite like the idea of making the movie seriously. The story of the transformers, while hokey, has a lot of depth. The trailer looks promising.

Unfortunately, people who've seen it tend to say it doesn't work, that cool effects try to carry the movie, but don't.

Stardust by Neil Gaiman

Ooh. I liked Stardust a lot more the second time I read it. I feel it could work well as a movie; there's not much *plot* but a lot of exciting things and character interaction and sparkly impressive things happen. And Gaiman seems excited, so you assume you got it right.

I'd *like* to see American Gods, but that probably has a heft more appropriate to an opera than a few hours of movie :)

Good Omens would be *amazing* but ever so difficult to get right.

Hellboy 2

OK, now Guillermo del Toro directed Pan's Labyrinth, am I allowed to think Hellboy has artistic merit? Actually, it was just a classic superhero film, though I tend to like those. But I love Hellboy and Abe, and the Nazi mysticism is ominous rather than ridiculous. I do hope the sequel is good, I could see it being entirely empty, or more fleshed than the first.

Poincare Theorem

Some of you may remember me going "aha" at Pizza a few weeks ago. Well, I was half right.

I was considering the Euler formula. For a 3d shape, Vertices + Faces = Edges + 2. Or with g holes through it, V+F=E+2-2g. (Thanks for Tom for helping visualise this.) (This is the Poincare Formula, well, one of them; like Euler, he had LOTS of formulas)

But isn't that "2" unaesthetic? Why 2? It should add to zero, surely?

After all, look at the 2d case. For a poly*gon* in a plane, vertices-edges=0. And for a polygon with a hole, also known as two polygons, vertices-edges=0 still.

But wait, that polygon is essentially a simple graph, drawn on a plane. But a plane can equally well be the surface of a sphere. And it's generally right to think in those terms because you avoid all the pesky infinities.

*Now* that looks suspiciously like what we had before. Our polygon has two faces: an area inside, and an area outside. Both are convex (one can be contracted to the centre, one to a point opposite the centre).

(NB: Everything has to be convex or it goes wrong. There are also some subtleties about when to have null points. I'll ignore this. The rule of thumb is that if you can have an extra vertex, line, or face, you do. This also eliminates non-convex faces, since you cut them into bits, and find the original answer was wrong.)

We can pull the same trick again in both directions. Look at a polyhedron. It has two *volumes*, an inside, and an outside. (The outside is convex if you consider it all connected to a point at infinity, ie. the back of a hypersphere.) This exactly cancels out the 2.

And if you add holes? The vertices and edges round the edges, and edges and faces inside all cancel out. But you make two volumes (the inside and the outside of the polyhedron) and two faces (the ones that had the holes formed inside them) non-convex. These can be fixed by dividing each into two, by adding two faces and two edges, which cancel.

Or look at a polygon. Ie. a series of points and edges filling a circle. The 1-d equivalent of a circle is two opposite points. (Honest, think about it. {x∈R: |x|=1}={-1,1}.) Thus to fill out a 1-d circle into a 2-d circle we also added two edges. For our 1-d circle we have the Euler formula v=2. Hard to argue with :)

But something seems to have gone wrong. Our 2s have come back in. Are we supposed to fill entirely the space we're considering, or imagine an n-d "circle" within it?

As it happens, it doesn't quite work like that. We've discovered http://mathworld.wolfram.com/PolyhedralFormula.html.

Define our shape in terms of simplices: a 0-d simplex is a point, a 1-d simplex is a line, a 2-d simplex is a triangle (topologically equivalent to any face), a 3-d simplex is a tetrahedron (topologically equivalent to a volume), etc.

Let n_k be the number of k-d simplexes. Let D be the dimension of surface we're trying to fill (eg. for a polyhedron, we're filling a 2-d surface, the surface of a sphere.)

Then n₀-n₀+n₀-n₀+... = 0 if D is odd, 2 if D is even.

You can consider more complex shapes by going up a dimension and thinking about the characteristic (chi=2-2g in 3d, this accounts for the shape of the surface you're filling).

Though even then self-intersecting shapes don't work.