## Hello World!

by

Wow,

Between teaching, research, paperwriting, and job applications, last semester was extremely busy for me. Obviously my blogging suffered (my last post was in July!), though I’m glad to see that Greg has been carrying the torch. With less to do this semester, I’m intending to start blogging again regularly. Hopefully my return will inspire Matt to come out of retirement!

Looking back at my old entries, I see that I put an enormous amount of effort into each one, leading to an inevitable decline in their frequency. Now that I’m back, I’d like to get more comfortable with writing posts that are short and friendly—less like Bourbaki and more like Isabel’s blog.

To emphasize my new commitment to casual blogging, I’ve decided to make my first post nothing more than a set of random links. Here goes:

1. The Harmonic Series Diverges Again and Again [PDF]: Twenty different proofs that the harmonic series diverges.
2. On Proof and Progress in Mathematics (arXiv), by William Thurston.
3. Hilbert’s Twenty-Fourth Problem [PDF]: This discusses the “lost” Hilbert problem, which is to formally define notions like homotopy between two proofs of the same theorem, and to decide by what criterion a given proof might be the “simplest possible”.
4. The Rise, Fall, and Possible Transfiguration of Triangle Geometry (JSTOR): A fascinating paper about an important subject in 19th-century mathematics which has almost completely ceased to exist.
5. Pythagoras’ Theorem for Areas (JSTOR): Let $A$, $B$, and $C$ be the areas of the three legs of a right tetrahedron, and let $D$ be the area of the hypotenuse. Then $A^2 + B^2 + C^2 = D^2$.
6. You Could Have Invented Spectral Sequences [PDF]: Possibly my favorite title for a math article, ever.
7. The Story of the 120-Cell [PDF]: An article about a certain regular four-dimensional polyhedron with 120 dodecahedral faces.

Enjoy!

### 6 Responses to “Hello World!”

1. Greg Muller Says:

Hooray!

Its good to have you back, Jim. I fully support your commitment to casual blogging, I try to stick to 2 hours writing time and 1000 words for my short lecture-style posts which seems to work pretty well.

I haven’t seem number 5 before, thats pretty nifty. Its reminds me of your frequent quest to find a proof of the Pythagorean theorem that generalizes to the higher dimensional versions. Have you had any luck with that?

2. Jim Belk Says:

I haven’t thought about it much recently. I’ve been intending to write a blog post about the multidimensional Pythagorean theorem for a while, although it’s sort of redundant given that the paper is online.

One point that the paper doesn’t make very clear is the connection between the multidimensional Pythagorean theorem and multilinear algebra. For example, the three-dimensional version is basically equivalent to the statement that the norm of the cross product of two vectors is equal to the area of the parallelogram that they span.

But no, I still don’t know of a proof of the Pythagorean Theorem that generalizes to the multidimensional case. I sure would like one.

3. Isabel Lugo Says:

I read the article “You Could Have Invented Spectral Sequences” when it was published in the Notices… I’m not convinced, although it’s quite a nice title.

But since you’ve implicitly called me the blogosphere’s anti-Bourbaki (a title I am honored to have, by the way), that’s not surprising.

4. Isabel Lugo Says:

Also, there seem to be a number of documents like your first link (on the harmonic series) which give many proofs of the same theorem. Yesterday I gave a link to a web page giving nineteen proofs of Euler’s theorem (that is, V – E + F = 2 for a planar graph) I may come up with a list of such lists of proofs.

5. Jim Belk Says: