Today I Learned: Generalised Pythagoras' Theorem

I had a hotpot dinner at church last Saturday. And speaking of hotpots - my friend broke her's (*). And no, that's neither a euphemism nor a metaphor. She was boiling water on the stove, forgot about it, only to come back to find a toasty dry pot roasting over blue flames. Quickly swiping the pot into the sink, she turned on the tap - only to find it shattered by the icy water.

This of course reminded me of my various childhood failures at drinking milk.

I used to drink about a litre of milk a day. I suppose that I should probably mention that China's experiments in pragmatic socialism had made this economically viable, but that's hardly the point. Instead, the point is that I was a champion of milk-drinking. So I was pretty shocked the first time that I tried milk here in Australia.

It was revolting.

It was all...creamy, and milky and not at all alike that watery malk produced by our proud proletariat Communist cows! Aussie milk: so gross.

So, to maintain my strong and healthy bones, I learned to take my milk with a big spoonful of sugar. Which again is slightly odd, because I was fine with taking medicine, in fact, I kind of liked it. But I'm getting distract when I should be distracting you with the story of that time that I mistook MSG for sugar - it certainly didn't help with my milk-loathing.

There was also one time when, home alone, I decided to microwave some milk during the commercial breaks (I can't remember the show, it was likely Sesame street or Play School), only to realise upon retrieving the glass that I'd forgotten to pour the actual milk into the glass (***). Shattered by my mindless folly, I took the glass to the sink and poured myself some water - only to find the glass shattered by this second act of mindless folly.

I've basically told you the same story twice: hot vessel + cold water = shattered pride. And this retelling of the same story but with minor details adjusted happens a lot in research mathematics. It is what we call generalising a result.

So let's talk about Pythagoras' theorem.

Oh wait, let's not - I've already done it here. Now there's an obvious generalisation to this result in higher dimensions. If we paraphrased/reinterpreted Pythagoras' theorem to say: the distance between two given points \((x_1,x_2)\) and \((y_1,y_2)\) on the plane \(\mathbb{R}^2\) is \begin{align}\sqrt{(y_1-x_1)^2+(y_2-x_2)^2},\end{align} then a fairly mundane generalisation would be that the distance between two points \((x_1,x_2,\ldots,x_n)\) and \((y_1,y_2,\ldots,y_n)\) on the plane \(\mathbb{R}^n\) is given by: \begin{align}\sqrt{(y_1-x_1)^2+(y_2-x_2)^2+\ldots+(y_n-x_n)^2}.\end{align}

But here's a more interesting generalisation that actually resembles the usual Pythagoras' theorem:

Take a square, slice off a small corner and you get a right-angled triangle. Take a cube, slice off a small corner and you get a right-angled tetrahedral. So take \(n\)-dimensional hypercube, slice off a small corner and you (should) get a right-angled \(n)-simplex (basically the generalisation of triangles and tetrahedrals)! Now, with a little imagination you should be able to see that this \(n\)-simplex has \(n+1\) "hyperfaces" - just like how a (\(2\)-dim) triangle has \(3\) sides or how a (\(3\)-dim) tetrahedral has \(4\) sides, You know how it is (****). And that one of these hyperfaces \(S_n\) is bigger than the rest \(S_0,\ldots,S_{n-1}\). Then we have following identity:\begin{align}vol(S_0)^2+\ldots+vol(S_{n-1})^2=vol(S_n)^2,\end{align} where \(vol(S_i)\) is the \((n-1)\)-dimensional volume of \(S_i\).

Awesome result, right? In particular, it says that the sum of the squares of the areas of the three smaller faces of a right-angled tetrahedral is equal to the square of the biggest face - i.e.: the sort of hypotenusoidal face.

To bad that the proof that I have for this isn't as cool as I want it to be. Although there are some nice ideas here, so very roughly outline it:

1. Show that a \(n\)-dimensional cone/pyramid with \((n-1)\)-dimensional base \(B\) has volume \(\frac{h}{n}vol(B)\), where \(h\) is the height of the apex (orthogonally) from the base. This can be done by considering about how the volume of \(B\) changes in the cross-sections of this cone parallel to the base, and then integrating these cross-sectional volumes along the height.
2. Conclude therefore that the volume of a right-angled simplex with vertices at \begin{align}(0,0,\ldots,0),(h_1,0,\ldots,0),(0,h_2,0,\ldots,0),\ldots,(0,\ldots,0,h_n)\end{align}is given by \(\frac{1}{n!}h_1\times h_2\times\ldots\times h_n\). And note that this may also be written as \(\frac{1}{n!}\) times the determinant of a matrix obtained by taking the non-origin vertex coordinates of our simplex as the rows of this matrix.
3. Combine this previous point with the fact that (high-dimensional versions of) shear mapping preserves volumes to get that the volume of a (general) \(n\)-simplex with one vertex at the origin, and the rest at \begin{align}(a_{1,1},a_{1,2},\ldots,a_{1,n}),(a_{2,1},a_{2,2},\ldots,a_{2,n}),\ldots,(a_{n,1},a_{n,2}\ldots,a_{n,n})\end{align} is given by \(\frac{1}{n!}\) times the determinant of the matrix \(A:=[a_{i,j}]\).
4. Now, embed this simplex in a one-dimension higher space, and in so doing + some shearing mappings, we get a formula for a \(n\)-simplex in \(\mathbb{R}^n\), which in term allows us to rearrage the vertex coordinate terms in this determinant into edge lengths. This is, to my current understanding, mainly (some impressive) algebraic manipulation, and may be found on page 124 and 125 of Sommerville's An Introduction to the Geometry of n Dimensions. It should be said that at this point, we're seeing that the square of the volume of our simplex is expressed completely in terms of the squares of the lengths of the edges of our simplex!
5. The proof of the last part that relates the volumes of all the hyperfaces of our simplex is also currently algebraic in nature, and may be found in the following paper.

Actually, in the process of writing this up, I've seen that Alvarez has an earlier proof of this result which seems much more elementary and uses induction. Maybe I should read it...

Maybe. =)

Update: yea, I read it, it's actually a much nicer proof. I mean, I don't get to make the cool observations about expressing volumes in terms of determinants and the shear mapping stuff, but it's a pretty chill induction proof.

*: I think that it was a hotpot pot, it might have been just a regular pot though. Who knows? She does, probably.

**: huh...it's curious now that I think about it...it's almost as if milk is actually an acquired taste, kinda like Vegemite?

***: on second thoughts, I could be lying here - I might have wanted water instead...huh, it was either water or milk...I'm leaning more towards water now.

****: this is an Arnold Schwarzenegger quote from Hercules in New York, but I just can't find footage of it on the interwebs. =(