Super nice video of the Hopf Fibration.

via trekearth

Have you guys heard of bitcoin yet? Here’s the original paper: Bitcoin – A Peer to Peer Electronic Cash System. Or immerse yourself in bitcoin culture at the bitcoin wiki and the bitcoin forum. Bitcoins are currently trading around $9 $18 (6/6) at the largest bitcoin market, mt. gox (featuring dark pools and soon offering margin trading and options). You can get hyperanalytical comparing various international markets at bitcoin charts. Or watch live bitcoin transactions with a block explorer and bitcoin monitor. Right now the bitcoin economy is pretty small (eg. stores, classifieds) but you can still gamble at the bitcoin poker room or bitcoin vegas poker. Or buy drugs! For now… You can donate to your favorite hackers. It’s all up to you! Here’s some hyperbolic commentary:

If you want to get started, download the bitcoin software to obtain a wallet and mine some bitcoins. Then, send me some at: 14gFPynCLAGmhhmZcPaaXPf6vBt7NKaY56

The Pythagorean Theorem is a true classic in the history of math. Generally, peeps credit the Ionian mathematician, Pythagoras, with the first proof of this theorem back around 500 BC. The oldest proof on wikipedia is Euclid’s proof and it looks pretty complicated. This is because Euclid was working within the classical framework of Euclidean geometry. As modern mathematicians, we have more powerful and diverse tools at our disposal for this proof. However, this makes the setup for our proof somewhat more complicated. Where do we begin? How much do we assume before beginning? I guess I’ll assume a basic knowledge of modern geometry. In particular, I’ll assume a basic knowledge of angles, lengths and triangles. The theorem concerns the length of the sides of a right triangle. Recall that the two sides connected to the right angle are called legs and the third side, opposite the right angle, is called the hypotenuse. Here’s a statement of the theorem:

Theorem: The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

So, according to the labels on the picture to the left, this relationship can be written as [pmath]a^2 + b^2 = c^2[/pmath]1. The easiest proof involves putting four of these right triangles together to form a square. You can do this in two ways, as demonstrated at the left. If you put them together in the top square, then you get the following proof:

Top Proof: The large square has side of length c and therefore it has area [pmath]A = c^2[/pmath]. However, we can see that this square is also composed of four triangles and a tiny square inside them. The area of each triangle is [pmath]1/2 ab[/pmath] and the area of the square is [pmath](b-a)^2[/pmath]. So, the area of the big square can also be written [pmath]4( 1/2 ab ) + (b-a)^2 [/pmath]. We can use algebra to rewrite this as [pmath]2ab + b^2 – 2ab + a^2 = b^2 + a^2[/pmath]. Since we already discovered that the area of the big square was [pmath]c^2[/pmath], we deduce that [pmath]a^2 + b^2 = c^2[/pmath]. The sum of the squares of the legs is equal to the square of the hypotenuse.

The argument for the bottom construction is similar. Both of these are found on wikipedia, along with several other proofs if you’re interested. My favorite might be the following proof by animated gif:

  1. I got pmath to work! And boy does it look crappy.

I’ve barely begun to blog about math and I’m already having typesetting nightmares. Mathematicians generally use tex or latex to typeset. For example, my thesis was a minor latex project involving  various extension/plugin packages for tricky typesetting (eg. amscd for commutative diagrams). I’m writing this blog with wordpress and so far I haven’t found a good typesetting plugin1. I’ve installed WP Math Publisher, but I’ve found it lacking. My main issue is that it doesn’t work, even after chmoding my way around some minor permission errors. I wound up borrowing this from wikipedia for the natural numbers post. But, even if I get WP Math Publisher to work, I noticed that WP Math Publisher might not have a , so I’m probably screwed when I get to the rational numbers. Ad hoc I could use textogif to typeset individual expressions2 into gifs, but that feels inelegant and laborious. Maybe, if I’m feeling ambitious, I’ll write my own wordpress plugin based on latex and textogif. If I’m really lucky, then I’ll get my dream job, where I’ll have some interested  and motivated chump student write this plugin for me. In the meantime, if you have any advice, hollllllla in the comments.

  1. I did find this plugin for footnotes, but I’m not sure if I like it.
  2. I love this example of a non-expression:

via waxinandmilkin

I am a wikipedia warrior. I love to read the pages. I’m hoping that wikipedia eventually replaces (overpriced) textbooks. I rewrote the alternating series page with gusto. My favorite discoveries since starting this blog are the chimera (genetic) and rational ignorance pages. My favorite quote is naturally from the empty set page:

“Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness.”

As a vegetarian, I guess two slices of white bread is better than eternal happiness. If you’re hungry, please take the swine.

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