The related concepts of Lagrange and Markov spectra have intrigued me since my student years; nowadays I study them using geometric methods in homogeneous dynamics, a strategy with roots in the 50s and 60s starting with W.Schmidt and brought to prominence by Dani, Margulis, Kazhdan and others.
I had the good fortune to be invited for lectures at Tata by Dani and Yale by Margulis, and I hope one day to reciprocate their kindness by proving something really strong about the objects they nurtured (something I have yet to achieve).
Edit: Unfortunately the Wikipedia page is very low on details about these beautiful subjects; if I find a nice, accessible survey, I will link it in a comment.
I've never been able to perceive the beauty that others such as yourself can see. Genuine question, where do you see the beauty? All I see in your link is a wall of equations, although I have a maths background it's all abstract and I can't get anything out of it.
Give not that which is holy unto the dogs, neither cast ye your pearls before swine, lest they trample them under their feet, and turn again and rend you.
You could say the same thing about music notation; the beauty is not in the notation, but the objects the equations describe. In order to perceive the beauty, you have to study the objects, not the notation.
Others (mathematicians) can peer through the notation to the thing beneath. That's extremely difficult for me to do, and once I've done it I can see something interesting, but beautiful? No, that tree is intriguing but beautiful to me it ain't.
I’ve stumbled across bits of program code I’d call beautiful: simple, understandable, clearly correct, and… elegant. Win I find them, I experience the same feelings as when seeing a beautiful painting. My wife’s like “why are you smiling and laughing at words on the screen”, but it’s not the words, it’s what they’re saying and how they’re saying it.
That must be what a mathematician feels when they see a beautiful proof, or formula, or relation.
There's lots of reasons that people find results in math beautiful, but to me there's some things that they seem to have in common, and it's similar to why people find songs beautiful -- surprise, inevitability, and relevance, and sometimes simplicity.
Statements that are obvious usually aren't interesting and aren't beautiful. (though there may be some like e^(i*pi) = 1 which are obvious consequences of the definition of the complex exponential function that people seem to like -- to me I think the definition of complex exponentiation is the beautiful part) The result in some sense should be counter-intuitive -- it had to have taken a spark of genius to see it.
Once you see it, it should seem confusing at first, but the proof of it should cause you to say "of course, this is the only way it could have been."
Finally, once you know about it, you immediately see useful consequences of it that open all new lanes of inquiry, or practical applications for it.
edit: I think also "simplicity" is a core part of it in that it distills a complex set of ideas to a concise statement -- e=mc^2, for example, or the example above.
I'm not sure about most people, but it is quite pretty how we start with (1, 1, 1) and x^2 + y^2 + z^2 = 3xyz which is all nice and simple then ... kaboom. Suddenly there are patterns everywhere. Not just similar patterns either, but a big range of things. Setting numbers to constants gets other known sequences like Fibonacci numbers, meaningful patterns when permuting, meaningful patterns when ordering, links to trees, links to irrational numbers, links to linear transforms. And all this is going on while many basic-sounding properties are apparently hard to analyse.
There is a lot going on for summing 3 squares. It is fun and unexpected. All the weirder because they don't seem to have linked all this to pi, e or i yet in an obvious way looking at the wiki page.
I see, sad those potential edits will be missed from appearing in the article, still could I ask you anyways to expand about "landmark results, open problems" ?
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[ 3.2 ms ] story [ 52.0 ms ] threadI had the good fortune to be invited for lectures at Tata by Dani and Yale by Margulis, and I hope one day to reciprocate their kindness by proving something really strong about the objects they nurtured (something I have yet to achieve).
Edit: Unfortunately the Wikipedia page is very low on details about these beautiful subjects; if I find a nice, accessible survey, I will link it in a comment.
https://en.wikipedia.org/wiki/Markov_spectrum
It also takes a mighty big dog to weigh a ton.
That must be what a mathematician feels when they see a beautiful proof, or formula, or relation.
Statements that are obvious usually aren't interesting and aren't beautiful. (though there may be some like e^(i*pi) = 1 which are obvious consequences of the definition of the complex exponential function that people seem to like -- to me I think the definition of complex exponentiation is the beautiful part) The result in some sense should be counter-intuitive -- it had to have taken a spark of genius to see it.
Once you see it, it should seem confusing at first, but the proof of it should cause you to say "of course, this is the only way it could have been."
Finally, once you know about it, you immediately see useful consequences of it that open all new lanes of inquiry, or practical applications for it.
edit: I think also "simplicity" is a core part of it in that it distills a complex set of ideas to a concise statement -- e=mc^2, for example, or the example above.
There is a lot going on for summing 3 squares. It is fun and unexpected. All the weirder because they don't seem to have linked all this to pi, e or i yet in an obvious way looking at the wiki page.
> Wikipedia can be edited by anyone
Tried once; never again.