## Monday, 4 May 2009

### Even the Science Magazines Can Blow It Sometimes

It is an in-joke among math types....
"What do you call a person who can't tell a coffee cup from a doughnut?"

... Yesterday I got the following release from Nature.com on my news link...

"A team of three has solved a 45-year-old problem in the mathematics of topology.

The Kervaire invariant problem is 'one of the major outstanding problems in algebraic and geometric topology' says fellow mathematician Nick Kuhn, at the University of Virginia in Charlottesville."

Ok, I admit, I never heard of the Kervaire problem, but that doesn't mean I am topologically stupid, and so when they showed the picture at the top, my mind jerked.. ????

Sure enough, later in the article, they had the comparison I expected, right before a rather complicated description of the details of the Kervair invariant. It said, "Algebraic topology is a way of describing the properties that objects with the same topology have in common. Topologically equivalent objects are objects that can be converted into each other by deforming but not tearing them: a sphere and an eggshell, for example, or a doughnut and a coffee cup." (my emphasis).

But of course they mean a coffee cup with a handle, like this one, so that both shapes are really a form of torus (doughnut) like objects. Spheres punctured with a single hole.

Ok, not they probably told some intern to get a file picture of a doughnut and a cup of coffee, and he did, but from top science mags... you just expect a little more.

Which brings me to my final question for someone who really knows topology.... is a Mobius Strip also a Genus one object like a torus?

vlorbik said...

i'll venture a guess: no.
"genus" would appear to apply to
surfaces bounding a solid.
now i'll probably have to look it up...

w'edia sez no on accounta orientability.

dricheson said...

A common definition of genus is that it is the number of nonintersecting closed curve along which we can cut a surface and not disconnect it.

Thus it is possible to apply it to nonorientable surfaces and surfaces with boundary. (And so the genus of a MÃ¶bius band is 1.)

However, in practice most people think of the genus as an invariant for orientable surfaces—tori or tori with disks removed.

Trashman said...

I find it amusing that if the bite in the doughnut was actually large enough to intersect the hole then the two would again be topologically equivalent (to spheres). Correct?

Pat B said...

Well, I like the idea of genus one... (which is almost the same as a proof in MY classroom..)..

so a mobius strip and a doughnut are both genus one...Ok, new joke, what do you call a guy who can't tell his mobius strip from a doughnut... ... ahhh, not funny, but thanks for the info anyway..

John said...

Mobius band has genus 1. Torus has genus 1. The difference is that the torus is orientable, the mobius band is nonorientable.

I think.

Paul said...

The nonorientable analogue of a torus is the (real) projective plane. In other words, a compact orientable surface of genus one is a torus, while a compact nonorientable surface of genus one is a projective plane. Your terminology may vary.