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gjm comments on [Open Thread] Stupid Questions (2014-02-17) - Less Wrong
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Which are the most beautiful mathematical or physical equations?
I am interested in the elegance of the equation itself (not its visualization, e.g. the Mandelbrot set). Yeah, I know that there is a difference in opinions, but I hope there will be some correlation among experts. I would like to have, let's say, 5 candidates.
This is a specific example, and a very good candidate, IMHO:
e^(i π) + 1 = 0
I would like to have about 5 equations, not just one, so even if you agree with my choice, please post even the equations that seem a bit less impressive, but they still have some kind of beauty in them.
Okay, just to give you another anchor, this one is nice too, but much more simple:
a^2 - b^2 = (a + b).(a - b)
Are you interested in elegance given physical relevance? For instance, Maxwell's equations aren't anything all that special in themselves, but when you discover that they explain approximately all of electricity, magnetism and light it's a different matter.
I didn't think about this part, because originally I was only thinking about math. Thank you for giving specific examples, so I can test my intuition to them.
Uhm... I'd say that "Δ . B = 0" doesn't trigger my feeling of awesomeness, however important it may be. Even "E = m * c^2" doesn't, and that has a lot of applause light connected with it. They are just too simple; they feel like "a + b = c".
On the other hand, I realized that if I don't know what the equation means, I can't decide whether it is good enough. So the meaning is a part of the utility function, but mathematical elegance is another part (the feeling of "oh, this is really equal to that?" like when the pieces of puzzle suddenly fit together) -- and I want the equations that satisfy both criteria.
Squark's examples are already great enough and probably all I need, so if you had some specific examples in mind, please post them here, but otherwise I already have what I wanted.