Fantastic article that explains the math (and physics) very clearly.
Ramanujan Surprises Again (2015)
31–40 of 96 posts
Re: Ramanujan Surprises Again (2015)
#32Earlier quoted context omitted.
What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.
i recall him explaining several shortcuts one can use to solve problems in seemingly impossible speeds by drawing on a breadth of experience from similar problems that you have memorized or are easy to compute and interpolating. its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.
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Re: Ramanujan Surprises Again (2015)
#33An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778 . It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number. (Technically, it works for any number of d…
In fact, there seems to be a lot of interesting things about 1729: https://en.wikipedia.org/wiki/1729_(number)
I love how the article starts with the most boring facts about 1729:
> 1729 is the natural number following 1728 and preceding 1730.
Re: Ramanujan Surprises Again (2015)
#34Don't we think that the credit for the number 1729 should belong to Hardy, for he took the cab and mentioned that number to Ramanujan. Of course, Ramanujan could see beauty in every number and would have produced something equally beautiful for some other number Hardy could utter.
Re: Ramanujan Surprises Again (2015)
#35Only neckbeards care about Ramanujan.
Re: Ramanujan Surprises Again (2015)
#36Earlier quoted context omitted.
Mathematics are the best expression of the transcendental divine. Pythagoras and Plato had the same perspective.
Funny to use the word transcendental there, since the Pythagoreans held ratios to be divine but couldn't figure out irrational numbers, like pi. They had trouble squaring that circle.
That's why I think Pythagoras refused to write down his doctrines. He knew there was more to be empirically discovered -- and he was wary of how text could become dogma. The divine he uncovered was based on a mathematical, harmonious cosmos; but he recognized it was beyond understanding in a lifetime. That's why Pythagorean mysticism is compatible with modern science -- he didn't write anything down!
2000 years later, Kepler had faith in a harmonious cosmos, and charged his model of harmony so it could fit the evidence. He elipsed the circles, instead of squaring them.
Fun fact #1: it is impossible to square a circle [1]
Fun fact #2: the Pythagoreans conducted the first attested scientific experiment in Western history (according to a recent PhD thesis at UMich [2])
Re: Ramanujan Surprises Again (2015)
#37Earlier quoted context omitted.
In fact, there seems to be a lot of interesting things about 1729: https://en.wikipedia.org/wiki/1729_(number)
I love how the article starts with the most boring facts about 1729: > 1729 is the natural number following 1728 and preceding 1730.
Heh. I've been reading HN for long enough to never be surprised by the capability of incredibly pedantic people to be incredibly pedantic.
Re: Ramanujan Surprises Again (2015)
#38Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan's investigations of Fermat's last theorem.
A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…
Feynman was undoubtedly a genius, but he also suffered from a need to be admired. The safecracking episodes at Los Alamos are a perfect example - giving the impression he was an expert safe cracker when his real methodology was guesswork and sometimes subterfuge (birthdays, anniversaries, or even subtlety observing someone inputting their combination).
Re: Ramanujan Surprises Again (2015)
#39An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778 . It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number. (Technically, it works for any number of d…
It doesn't appear to be anything fundamental, just a number that makes the proof technique work out. As the authors point out, that dimension is only a condition to get some loose bound working, not a necessary condition generally:
It is possible to improve the factor K = 1728 appearing in Corollary 5.5, at the expense of introducing various technical complications into the algorithm. In this section we outline a number of such modifications that together reduce the constant to K = 8 + ϵ, so that the modified algorithm achieves M(n) = O(n log n) for any d ≥ 9 (rather than d ≥ 1729).
Re: Ramanujan Surprises Again (2015)
#40Earlier quoted context omitted.
A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…
I think most of us are impressed by computational parlor tricks (and indeed raw computational intelligence in general -- being able to process information and compute quickly and accurately), but for me, genius goes beyond that. Genius is about having rare and useful insights that the rest of us are incapable of, and that a computer is unable to easily replicate. For instance, there was this thing on Twitter recently…
It never occurred to me that 7% of 50 is equal to 50% of 7 perhaps because they are equally easy to calculate? Multiply 7 by 5 and fix the decimal point.