By 苏剑林 | March 14, 2019
Today is March 14, which happens to be 3.14—the "Pi Day" ( $\pi$ day) that many science students love to joke about~
Pi Day
There are many stories to tell about $\pi$. Using the "most beautiful formula" $e^{i\pi}+1=0$ to tell the story of $\pi$ seems a bit too cliché and lacks a bit of technical depth. However, I believe that the series of formulas produced by the "genius" mathematician Ramanujan will never go out of style, for example:
$$\sqrt{\phi +2}-\phi =\frac{e^{{-{\frac{2\pi }{5}}}}}{1+{\frac{e^{{-2\pi }}}{1+{\frac{e^{{-4\pi }}}{1+{\frac{e^{{-6\pi }}}{1+\,\cdots }}}}}}}=0.2840...,\quad \phi=\frac{1+\sqrt{5}}{2}$$
Look, Euler's formula connecting $e,i,\pi,1,0$ is great, but mine connects $e, \pi$, and the golden ratio together in the form of an infinite continued fraction!
Another example:
$$\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum_{k=0}^{\infty}\frac{(4k)!}{(k!)^4}\frac{1103+26390k}{396^{4k}}$$
Isn't it just a series for $\pi$? What's the big deal? The big deal is that if you take only the first term of this series, you can get $\pi=3.1415927...$. The very first term gives 8 significant digits. If you analyze it a bit more, you will find that this series converges incredibly fast... In fact, it (and its variants) is the foundational formula currently used by computers to calculate billions of digits of Pi. (More details can be found on Wikipedia)
Later generations derived even more advanced versions:
$$\frac { 1 } { \pi } = \frac { 12 } { ( 640320 ) ^ { 3 / 2 } } \sum _ { k = 0 } ^ { \infty } \frac { ( 6 k ) ! ( 545140134 k + 13591409 ) } { ( 3 k ) ! ( k ! ) ^ { 3 } ( - 262537412640768000 ) ^ { k } }$$
Its theoretical basis is:
$$e^{\pi \sqrt{163}} \approx 640320^{3} + 744$$
By the way, $e^{\pi \sqrt{163}}$ is also known as the "Ramanujan constant."
There are many more of Ramanujan's spectacular formulas (see "How were those spectacular formulas of Ramanujan discovered?"). It feels like they could only be produced by "cheating." While many formulas might not have practical utility, many of them far exceed our imagination—and "our" refers not just to ordinary people, but also to many great mathematicians. For instance, Hardy commented upon seeing these formulas:
They must be true because, if they were not true, no one would have had the imagination to invent them.
You have to realize that coming up with a new formula for $\pi$ or $e$ is often local-hero work, but Ramanujan's formulas usually blend $\pi, e$, etc., in very bizarre ways using series, square roots, and continued fractions, and yet they are somehow correct! So it's no wonder Hardy spoke this way; ordinary people wouldn't even dare to imagine such formulas.
How about one more?
\begin{aligned}R_n^{+}:=& \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\Large {2^{n+1}}]{x^2 + \ln^2\!\cos x} \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\cdots+\frac{1}{2}\sqrt{ \frac{1}{2}+ \frac{1}{2}\sqrt{ \frac{\ln^{2}\!\cos x}{ x^2 + \ln^2\! \cos x}}}}}\,\mathrm{d}x\\ R_n^{-}:=& \frac{2}{\pi}\int_{0}^{\pi/2}\frac{1}{\sqrt[\Large {2^{n+1}}]{x^2 + \ln^2\!\cos x}} \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\cdots+\frac{1}{2}\sqrt{ \frac{1}{2}+ \frac{1}{2}\sqrt{ \frac{\ln^{2}\!\cos x}{ x^2 + \ln^2\! \cos x}}}}}\,\mathrm{d}x. \end{aligned}
Then we have $R_n^{+}= \sqrt[\Large {2^n}]{\ln 2}$ and $R_n^{-}= \frac{1}{\sqrt[\Large {2^n}]{\ln 2}}$. Could you imagine such complex integrals yielding such simple results? (See "Ramanujan Log-Trigonometric Integrals")
Ten Years of Blogging
By the way, there is another thing. After entering March, Scientific Spaces officially enters its tenth year. This means I have been blogging for ten years.
In fact, I don't remember exactly which day I started blogging, but it should have been in March 2009. Since today is Pi Day, let's treat today as the anniversary~
I first encountered the internet in 2006. For the first three years, I was passionate about running IT forums (if interested, see this article). Later, the craze passed and IT forums gradually declined. For some other reasons as well, I focused on writing a blog instead, as a personal space is easier to manage.
At first, it was mainly popular science. A blog that was very popular back then was the "Songshuhui (Association of Science Squirrels)", and I was mainly inspired by them. Slowly, I focused on recording my own studies and research. The theme of Scientific Spaces has changed along with my interests—starting with astronomy, then mathematics and physics, and now mostly machine learning. That's why it has formed its current "mishmash" of ten categories~
In the past two years, perhaps because I have written more on machine learning and kept up with the times, the blog's popularity has increased. Currently, I average about one article per week, and they are basically all original content. I hope to explain the principles of a certain topic clearly using accessible language. Although the themes of the articles change, I still haven't forgotten the original philosophy when I started the blog: "Cracking the nuts of science, making science popular." I hope to continue sharing and moving forward with everyone.
Ten years is not a short cycle. In these ten years, although many things have become passing visitors, many websites have persisted. For instance, the aforementioned Songshuhui is still updated and has developed platforms like "Guokr.com"; then there is the "Mathematics Development Forum", which has also existed for over ten years and remains popular—it wouldn't be an exaggeration to say it is currently the highest-quality mathematics forum. There are other websites too; I can't recall them all at the moment, so I won't list them. Scientific Spaces has just reached its first decade; I hope to walk many more decades with you all.
Ten years later, we are friends and can still say hello~