Why are some equations considered beautiful in mathematics?

[JayJohn85]

What does it exactly mean if a equation is elegance and beautiful? Can you post some examples and your reason for why you find it beautiful?

 

[WannabeNewton]

Elegance and beauty are extremely subjective but it is not inconceivable that one would find an equation of physics that was both concise and far reaching in consequence and application to be of a beautiful character. My go to examples are Maxwell's equations recast using differential forms which take the form $dF = 0, d\star F = 4\pi \star j$. Obviously these are extremely concise and we know of course that Maxwell's equations are far reaching in scope in that they explain the dynamics of the classical electromagnetic field. That these equations codify the nature of this field is quite breathtaking when you look at just how elegant they look.

 

[A.T.]

What does it exactly mean if a equation is elegance and beautiful? Can you post some examples and your reason for why you find it beautiful?

Simple yet generally applicable or relating seemingly unrelated quantities. One example:
http://en.wikipedia.org/wiki/Euler's_identity

 

[micromass]

My favorite is

$$\prod_{p~\text{prime}} \frac{1}{1-p^{-2}} = \frac{\pi^2}{6}$$

It's nice because it relates two entirely different quantities, namely prime numbers which arise in number theory, and $\pi$ which is a geometric concept. So it relates two very different fields of mathematics.

I don't like the $e^{i\pi} + 1 =0$ very much, because it is essentially a definition to me. The equation above is much deeper. You can actually check it by entering the LHS in a calculator (or at least, a partial product), and you'll get an approximation to $\pi$. So I think this is something deeper than Euler's formula.

 

[dx]

My favorite is

$$\prod_{p~\text{prime}} \frac{1}{1-p^{-2}} = \frac{\pi^2}{6}$$

It's nice because it relates two entirely different quantities, namely prime numbers which arise in number theory, and $\pi$ which is a geometric concept. So it relates two very different fields of mathematics.

A related one:

$$\prod_{p~\text{prime}} \frac{1}{1-p^{-s}} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \dots$$

 

[micromass]

A related one:

$$\prod_{p~\text{prime}} \frac{1}{1-p^{-s}} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \dots$$

This also shows that there are infinitely many primes, just take $s=1$.

 

[WannabeNewton]

On a similar note, proofs can also be seen as extremely beautiful and elegant (in my opinion a truly elegant proof is much more captivating than an equation of the same caliber). I refer to the two archetypal examples of an elegant / beautiful proof: the proof of the uncountability of the reals via the diagonal argument as proved by Cantor and Cantor's proof that the cardinality of the power set of a set is strictly greater than that of the set itself. See here for the latter: http://www.math.ucla.edu/~hbe/resource/general/131a.3.06w/cantor.pdf and see here for the former: http://planetmath.org/cantorsdiagonalargument

In the spirit of the above posts on the infinitude of the primes, see here for an elegant / beautiful topological proof of the result: http://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes

 

[colliflour]

It's my opinion that dimensionless groups are the most beautiful. I.e. the ones that are a single parameter generated by combining other parameters of a given scenario which when substituted into some governing equation for a physical process, dictate the the qualitative behavior based on whether some critical value of the dimensionless group is reached. The obvious example is Reynolds number for turbulence in flow systems, but there are many others.

 

[JayJohn85]

Thank you for the interesting replies.

 

[Number Nine]

One of the most beautiful identities in algebra.
Let $G$ and $H$ be groups and let $f: G \rightarrow H$ be a homomorphism:
$$G \diagup ker(f) = f(G)$$