1. Aug 30, 2014

### micromass

A theorem in mathematics can be beautiful in a lot of different ways. The result can be surprising or satisfying. Or perhaps the proof is very elegant and beautiful. Or maybe the theorem nice because it can be applied to other mathematics, or physics or engineering, or anything.

So give here the theorems you find especially beautiful and be sure to say why you like it so much. There is no wrong answer!

2. Aug 30, 2014

### jhae2.718

I am particularly fond of the Cayley transformation,\begin{align*}C&=(1-Q)(1+Q)^{-1} = (1+Q)^{-1}(1-Q)\\Q&=(1-C)(1+C)^{-1} = (1+C)^{-1}(1-C)\end{align*}where $C$ is an orthogonal matrix and $Q$ is a skew-symmetric matrix.

Why?
1. The right hand side commutes.
2. The forward transformation is the same form as the reverse; you just need to swap $C$ and $Q$.
3. It is an extremely useful result for passively parameterizing rigid-body attitude.
4. It can be used to transform rotation into a cross-product operation.

Cayley, Arthur. "Sur quelques propriétés des déterminants gauches." Journal für die reine und angewandte Mathematik 32 (1846): 119-123.

3. Aug 30, 2014

### micromass

And even more interesting: it works in infinite dimensions too. You can use it to prove a very general spectral theorem.

4. Aug 31, 2014

### Staff: Mentor

I'm a physics buff. I particularly love Noether's Theorem and Liouville's Theorem because they have immense importance in physics, yet they were derived in pure math ignorant of the physical significance.

I also delight in the Schrödinger equation because it ties knowledge to energy. There can be no knowledge without energy expenditure. That is stunning.

5. Aug 31, 2014

### PeroK

I have always liked:

1. The Remainder Theorem
2. The Binomial Theorem
3. The Bolzano-Weierstrass Theorem

It seems that so much rests on them. They are like perfect foundations.

6. Aug 31, 2014

### wukunlin

The squeeze theorem.

For some reason it reminds me of piping bags
http://www.papstar-products.com/papstar_pe/prodpic/100-Piping-bag-2-6-l-55-cm-x-26-5-cm-transparent-12488_b_0.JPG [Broken]

Last edited by a moderator: May 6, 2017
7. Aug 31, 2014

### leroyjenkens

Pythagorean theorem. Not only because it has helped me so much with problems I was probably supposed to solve in a more difficult manner, but also because it's the only mathematical theorem I can think of.

8. Aug 31, 2014

### micromass

The Pythagorean theorem is indeed one of those theorems which are extremely beautiful. The result is totally unexpected in my opinion: I see no a priori reason or explanation for why the sides of a rectangular triangle should behave in this way. The proofs of the theorem are also quite nice. There are a lot of proofs too (not all of which are that rigorous).

http://www.cut-the-knot.org/pythagoras/

9. Aug 31, 2014

### Tosh5457

Last edited by a moderator: May 6, 2017
10. Aug 31, 2014

### BOAS

I don't know if it has a nice name, but it was the first theorem I worked through largely on my own (had done others in class), and the feeling of satisfaction was great.

"The sum of the first n positive integers is $$\frac{n(n+1)}{2}$$"

11. Aug 31, 2014

### micromass

Stokes' Theorem is extremely elegant and very important. It has a lot of consequences such as Green's theorem, the divergence theorem, the fundamental theorem of algebra, Brouwer's fixed point theorem, the invariance of domain theorem, a lot of complex analysis theorems, etc. It's definitely one of the most important theorems out there. Too bad that most books don't give a nice derivation of the theorem.

The residue theorem is also very nice. Complex analysis has a lot of beautiful theorems. For exampel, the Cauchy integral formula is very nice too: http://en.wikipedia.org/wiki/Cauchy's_integral_formula

This was the theorem that Gauss discovered when he was about 5 years old:

Another very nice proof of this result is geometrical: http://www.mathsisfun.com/algebra/triangular-numbers.html

The generalization of this result would be: what is the sum

$$1^2 + 2^2 + 3^2 + .... + n^2$$

or in general, what is the sum of $$1^k + 2^k + 3^k + ... + n^k$$

This is not at all obvious. The results for the first few $k$ are:

$$\begin{eqnarray*} 1 + 2 + 3 + 4 + 5 + ... + n & = & \frac{1}{2}n^2 + \frac{1}{2}n\\ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + ... + n^2 & = & \frac{1}{3}n^3 + \frac{1}{2}n^2 + \frac{1}{6}n\\ 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + ... + n^3 & = & \frac{1}{4}n^4 + \frac{1}{2}n^3 + \frac{1}{4}n^2\\ 1^4 + 2^4 + 3^4 + 4^4 + 5^4 + ... + n^4 & = & \frac{1}{5}n^5 + \frac{1}{2}n^4 + \frac{1}{3}n^3 - \frac{1}{30}n\\ 1^5 + 2^5 + 3^5 + 4^5 + 5^5 + ... + n^5 & = & \frac{1}{6}n^6 + \frac{1}{2}n^5 + \frac{5}{12}n^5 - \frac{1}{12}n^2\\ 1^6 + 2^6 + 3^6 + 4^6 + 5^6 + ... + n^6 & = & \frac{1}{7}n^7 + \frac{1}{2}n^6 + \frac{1}{2}n^5 - \frac{1}{6}n^3 + \frac{1}{42}n\\ 1^7 + 2^7 + 3^7 + 4^7 + 5^7 + ... + n^7 & = & \frac{1}{8}n^8 + \frac{1}{2}n^7 + \frac{7}{12}n^6 - \frac{7}{24}n^4 + \frac{1}{12}n^2\\ 1^8 + 2^8 + 3^8 + 4^8 + 5^8 + ... + n^8 & = & \frac{1}{9}n^9 + \frac{1}{2}n^8 + \frac{2}{3}n^7 - \frac{7}{15}n^5 + \frac{2}{9}n^3 - \frac{1}{30}n\\ 1^9 + 2^9 + 3^9 + 4^9 + 5^9 + ...+ n^9 & = & \frac{1}{10}n^{10} + \frac{1}{2}n^9 + \frac{3}{4}n^8 - \frac{7}{10}n^6 + \frac{1}{2}n^4 - \frac{3}{20}n^2\\ 1^{10} + 2^{10} + 3^{10} + 4^{10}+ 5^{10} + ... + n^{10} & =& \frac{1}{11}n^{11} + \frac{1}{2}n^{10} + \frac{5}{6}n^9 - n^7 + n^5 - \frac{1}{12}n^3 + \frac{5}{66}n \end{eqnarray*}$$

The question is to find a pattern for the general case. We can see some parts of the pattern, but finding the general case is not at all easy. If you want a tough challenge, you can try it.

Here is the solution: http://en.wikipedia.org/wiki/Faulhaber's_formula
A very nice derivation of the general formula can be found in the following intriguing document which attempts to generalize calculus to discrete situations: https://www.cs.purdue.edu/homes/dgleich/publications/Gleich 2005 - finite calculus.pdf

Last edited by a moderator: May 6, 2017
12. Aug 31, 2014

### Matterwave

I was going to say Stoke's theorem...but it's been taken.

13. Aug 31, 2014

### micromass

***Stokes'

14. Aug 31, 2014

### Matterwave

Stoke's this just flows naturally from my keyboard.

15. Sep 1, 2014

### jostpuur

There are some truly horrendous "theorems" out there that do not even satisfy the usual standards for theorems. Examples include the Noether's theorem, Bloch's theorem and Spin-statistics theorem. I have never seen these being formulated in such form that they could be called mathematical theorems, but still everybody insists that they would be theorems. All the necessary definitions and assumptions are missing, and only the claims are present. If people don't know how to formulate ideas as theorems, then they should be called conjectures.

16. Sep 1, 2014

### Hepth

I never knew Noether's theorem was not a "theorem" in peoples eyes? Can you explain that? She was a mathematician, I had always assumed it was quite straight forward.
Heres a newer English translation :

http://arxiv.org/pdf/physics/0503066

17. Sep 1, 2014

### WannabeNewton

Don't take the bait my friend.

18. Sep 1, 2014

### Jorriss

One can also see Arnold's statement of Noether's Theorem in his classical mechanics text.

19. Sep 1, 2014

### micromass

Sure, there are some results in physics which are not mathematically rigorous (or are not presented as such). But Noether's theorem is a horrible example of this. See Arnolds book which gives a rigorous version of it.

20. Sep 1, 2014

### td21

Godel's incompleteness theorem.