Periodic Table of Shapes

In this article, mathematicians are creating a “Periodic Table of Shapes” in three, for, and five dimensions.

The shapes are Fano varieties, but I don’t really understand what that means. Apparently in two dimensions there are nine of them, and in four dimensions they estimate 500 million of them. About how many of them are there in three dimensions? What do they look like in two and in three dimensions? If they’re shapes, it ought to be possible to draw them, especially in 2 dimensions. What do these shapes look like? Are those four shapes thee-dimensional ones, or slices of four- or five-dimensional ones? If they’re three-dimensional ones, what makes them elemental shapes?

Since the number is finite in four dimensions, it presumably is also finite in three, so they can’t just mean “sphere, torus, two-holed surface, three-holed surface, etc.”

Googling, there are several other articles on this, but they all seem to have about the same level of information, and don’t answer the questions I have above.

Here’s another article with a couple more images.

Found one. The Clebsch Diagonal surface is a fairly standard embedding of the cubic surface into P[sup]3[/sup], which would make it third on your list of the nine two-dimensional varieties.

According to the article, the pictures are two-dimensional slices of much more elaborate three-dimensional shapes. I’m rapidly going out my depth here, so I don’t have a good answer for your last few questions, but it’s worth mentioning that these are surfaces that look like the projective plane, and would be embedded in projective space of a higher dimension. I don’t know if the number of handles is still a useful invariant in that situation.

Here’s a New Scientist article plus video that might be slightly more accessible. They’re still having a tough time with it, though, beyond saying “here it is.”

Fano varieties :smiley:

S’rusly though, that Wikipedia entry must take the prize for the most incomprehensible to anyone who does not have specialist technical knowledge of the subject already.

True. Here’s what they have to say on the subject:

What the hell does that mean?

“My anticanonical bundle is ample” sounds like something from the Hungarian dictionary.

But what if we have something that’s Kodaira dimension isn’t negative infinity? Is it a singular complete variety whose anticanonical bundle is ample? A non-singular incomplete variety whose anticanonical bundle is ample? A non-singular complete variety whose canonical bundle is ample? Or maybe a non-singular complete variety whose anticanonical bundle isn’t ample? The possibilities…

is it even **POSSIBLE **to show a 5 dimensional shape represented in 2 or 3 dimensions?

When I read things like this, I wonder if there isn’t some hoax going on within this small community of folks that claim to understand it. Since no one really understands it, it cannot be refuted and everyone just nods their heads, not wanting to look like the idiot of the group.

You know, I was thinking the same thing. :wink:

If even Nicolas Bourbaki doesn’t understand it, I don’t think any of us regular, individual people stand any chance at all.

Snifit writes:

> “My anticanonical bundle is ample” sounds like something from the Hungarian
> dictionary.

It’s a great pick-up line in mathematician bars: “Hey, babe, I bet your anticanonical bundle is ample. I bet you have a non-singular complete variety of them. To be precise, there seems to be two of them, and they are really complete.”

All I want to know is the nine 2D shapes. Those at least should be picturable. I have a hard time imagining nine shapes that every other shape can be made from. You can build all straight sided shapes with triangles, but even each triangle can be a different shape.

Thanks, these links are helping. The New Scientist article aswers one question, I thnk: 102 “atomic” 3D shapes were found in the 1980s, and they’re expecting a few thousand total, but don’t seem to have a final number yet.

The two-dimensional shapes aren’t curves in 2-space, as I was thinking, they’re 2D surfaces in 3-space (or 4-space, etc. if necessary), so visualizing the 3D ones would be a lot harder than I was expecting.

Lets concentrate on the 2D shapes, with the linked Clecsch Diagonal surface one of the nine. Some of the shapes have to be simple, right? Is there some kind of correspondence with the typical topological surfaces? Are a plane, a sphere, a torus, a Klein bottle, and a cross cap five others, with a two-holed surface maybe being a “molecule” of two tori? If this is the case, then it’s starting to make some sense.

There’s a link on the Cross Cap page to a page on Classification of closed surfaces, but there’s no indication of there being nine basic shapes.

Algebraic varieties are defined over some field k (usually algebraically closed) that’s not necessarily the usual fields of real or complex numbers. That’s what separates them from manifolds; you don’t have any useful notion of smooth, or even continuity, over an arbitrary field, and so the construction are much more complicated. There are analogues of, say, the cotangent bundle, but they’re trickier to define and use than for manifolds.

In dimension 2, the Fano varieties are P^1 x P^1 and P^2 with up to 8 blow-ups. Over ground field C, the first is S^2 x S^2, and the latter looks like a connected sum CP^2 # k(-CP^2) for k = 0, …, 8. (I don’t know how to make overlines here, so -CP^2 is supposed to denote CP^2 with the opposite orientation.) Why 8? Sorry, I don’t know. Those are exactly the smooth, compact, oriented 4-manifolds with complex structure and positive, constant Ricci curvature, so I’m sure that’s important somehow.

Word to the wise: you do NOT want to utter this phrase in a speakeasy in Bratislava.

Once a gypsy hooker decides you’ve proposed to her, her uncles will hound you for days, and you may have to chloroform a nun whose habit is your size, pick enough pockets to bribe your way aboard a tramp steamer at 2 am, and spend 36 hours in a cargo hold full of testy geese, Hot Budapest Paprika, and rats. Or so I’ve heard…

Not even close. That one doesn’t even have Wikipedia’s insufficient context tag. :smiley:
But yeah, I’m pretty much on your side. I’m a first year grad student with no knowledge of algebraic geometry other than what I’ve picked up via osmosis or from what my friends have told me. The wikipedia article that actually defines algebraic varieties is to me written in plain English; I would not have noticed there were specialized symbols or lingo unless they were specifically pointed out to me. The entry defining a Fano variety I do not understand at all, but I recognize the language they’re using as standard, and I feel like it would make perfect sense if I studied it for a year out of Hartshorne or Eisenbud. The article I linked up top, OTOH? What the hell . . .

Algebraic geometry isn’t my thing either, but I’ll try to take a stab at it. There’s a lot of complication definitions to parse through (which is why algebraic geometry isn’t my thing either). To simplify matters, let’s work over C. Then a Fano variety V is:

  • An algebraic variety: It’s defined as the set in some C^N (affine) or CP^N (projective) where some fixed polynomials f_1, …, f_n vanish. (There’s a more general and intrinsic definition, but it requires discussion of schemes and gets very complicated quickly.)
  • It’s non-singular: The matrix of partial derivatives of the f_i is non-singular. The f_i are just polynomials, so this determinant is well-defined over any (commutative) field. Geometrically, this condition means that V locally looks like a projection C^N = V + C^m -> V. Roughly speaking, there aren’t any corners or self-intersections.
  • Its anticanonical bundle…: A smooth manifold M has a cotangent bundle T*M, which is just the dual of the usual tangent bundle TM. This is a vector bundle of degree n = dim M; we can take its nth exterior power det(T^M), which is called the determinant bundle of M. (In fact, the transition functions on the bundle are given by the determinant of the ones for TM.) You can do the same thing with the cotangent bundle of an algebraic variety, though you have to be more careful about defining the cotangent bundle when you’re over an arbitrary field k. The nth exterior product of that bundle is called the canonical bundle, and its inverse is the anticanonical bundle. Line bundles form a group under tensor product; in terms of explicit transition functions, a bundle with transition functions {f} will have an inverse with transition functions {1/f}.
  • …is ample: A line bundle L -> V is very ample if there’s an embedding i into projective space P^N such that i^* O(1) = L, where O(1) is the inverse of the tautological bundle on P^N. The significance is that for any very ample bundle L and coherent sheaf F, the product F ensor L^n is finitely generated by global sections for sufficiently large n. (Coherent basically means that it’s locally generated by finitely many sections; the point here is that those sections can be made global). An ample bundle L is one such that L^n is very ample for sufficiently large n.

I can’t explain where the bit about the Kodaira dimension comes in the wikipedia article. I do know that simply-connected complex surfaces S are classified in part by their Kodaira dimension: If kod(S) = 1, then S is of elliptic type; if kod(S) = 0, then M is a K3 surface; if kod(S) = -\infty, then S is rational or ruled. (There’s another case, kod(S) = 2, which corresponds to “general type”; this is a condition on intersection numbers of curves with the canonical divisor.

My nipples explode with delight!

One thing to keep in mind is that these will all be complex varieties. So the “one dimensional projective line” is actually a 2 real-dimensional surface. But it is easy to visualize as a sphere with the north pole being the point at infinity. To see this, imagine the sphere sitting on the origin of the ordinary complex plane. To find the point on the sphere corresponding to a number in the plane, draw a line from the top of the sphere to the point on the plane. Such a line will intersect the sphere and that point of intersection is the point on the sphere that corresponds. Nothing corresponds to the north pole of course.

Now a 2 dimensional variety will have 4 real dimensions. Lot’s of luck trying to visualize that!

Hmmmm… None of the articles mentioned that these were complex shapes. Might have been nice to have been told that. But hey, they probably thought “If they want to know more, they can ask the 'Dope”.

So for 1D, you’d have lines in the complex plane = lines in a 2D plane. For 2D, you’d have shapes which are surfaces in 4D space. For 3D, you have shapes which are volumes in 6D. Something like that?

The Algebraic Varieties page is helpful, particularly the Examples.

Thanks everyone, I can kind of see where this is coming from.