Saturday, March 14, 2009

Happy Pi Day!

Pi fun fact:

As you may have learned in high school geometry (unless you went to my yeshiva), pi is defined as a circle's circumference (the distance around) divided by its diameter (the distance across). It is approximately equal to 3.14 (that's why Pi Day is 3/14).

Now imagine a circular disk. If we measure its circumference and diameter, we find that their ratio is pi. What if we start to spin the disk (for example, a CD in a CD player)? According to Einstein's Special Theory of Relativity, a moving object's length is shortened along its direction of motion as it moves (for example, if you throw a yard stick, its length becomes less than a yard while it is moving). Now, the outer edge of the disk is moving, thus its circumference is shortened. But the diameter remains unchanged, since it is not along the direction of motion. Thus the ratio of circumference to diameter is now less than pi!

What does this mean? It means that the geometry you learned in high school ("Euclidean Geometry") is not sufficient to describe relativistic motion. In fact, the spinning disk in free space is indistinguishable from a stationary disk in a gravitational field (that's Einstein's "equivalence principle") so this must mean that space under the influence of gravity is not describable by Euclidean geometry. In other words, space can be curved!

Yep kids, the universe is amazing, even without angels and witches.

Incidentally, I remember learning a gemara which said that the ratio of a circle's circumference to its diameter is 3 (I think it was talking about fitting the Torah into the holy ark). I guess you could say that the gemara was just being approximate (I think some meforshim say that), but I'd guess that Babylonian rabbis at the start of the dark ages just weren't math experts. Of course if you're a true believer then you'd say that the chazal knew everything there is to know, even though there is another gemara which says that the sun goes behind a curtain every night.

Edit: I've been informed that is today is merely American Pi Day. The British Pi Day does not occur until the 31st of April, which is never.

26 comments:

  1. "What does this mean? It means that the geometry you learned in high school ("Euclidean Geometry") is not sufficient to describe relativistic motion."

    Apikores, this is not the way it works. You don't need GR to resolve the Ehrenfest Paradox, since there is a kinematic solution within SR that works just fine. You can look up some of the papers by Gron on the subject.

    Out of curiosity, did you decide to pursue a science degree now that you're done with religion?

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  2. YS,

    You're probably right, I'm not really qualified to know. I was just regurgitating an argument I remember reading in a popular science book once (I can't remember which), in honor of pi day. I might have got some of the details wrong.

    I'm trying to remain anonymous, so I don't want to reveal too much about myself, but yes, I am pursuing a science degree, and I'm still at the very beginning of my studies. The only exposure I have had to GR is from reading a little about it myself. This post was just meant to be a little fun tidbit of relativity, not anything really precise.

    Do you have a blog? I'm always interested to know what someone with actual science knowledge thinks.

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  3. (Also, I just skimmed the Wikipedia article on the Ehrenfest paradox, and it seems to be a pretty complicated problem. I guess it would have been better for me to talk about the accelerating elevator, but that wouldn't have had anything to do with pi.)

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  4. No, I don't have a blog. I've toyed with the idea back when I had to while away hours in the lab waiting for for the reconstructions to finish, but I never really got around to doing it.

    The reason I was asking was because I thought we might be kindred spirit, as an ex-Lubavitcher atheist myself.

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  5. Btw, I was really disappointed to learn that Einstein's equivalence principle is really not all that it's cracked up to be. So, before you start posting on accelerating elevators, I would recommend reading Einsteins Mistakes by Hans Ohanian. The book presupposes a fair bit of physics knowledge, (like knowing how to manipulate a tensor), but it's still very readable.

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  6. Cool. I didn't know there were other ex-lubavitcher atheist scientists out there! Thanks for the book reference. I'm very interested to learn more about relativity, but I still have a lot of catching up to do. I'm afraid I don't have much experience manipulating tensors (anything more complicated than matrices is beyond me at this point), but I'll take a look at the book anyway if I can find it at my campus library.

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  7. Go read Eiruvin again. When the chachamim talk about eiruv techumin they come pretty close to Pi. Not exactly it but they use their number (I think it's something like 3.12) in the correct way.

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  8. So they got it right in one gemara and wrong in another. No big deal, since pi was known to the Greeks hundreds of years before. My point was that chazal weren't infallible and didn't know everything.

    In yeshiva, my rabbeim would try to explain away any apparent mistake chazal of a godol had made. They couldn't admit that these people were humans and only had access to the knowledge of their times.

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  9. There are multiple uses of Pi in the Talmud. Some of them use 22/7 which was an approximate that many ancients used in actual calculations (it works quite well. The reason it is a good approximation is related to the theory of continued fractions- http://en.wikipedia.org/wiki/Continued_fraction which are very cool).

    There are actually far more serious lapses in the Talmud about both math and science. There's a Gemmarah in Kilayim which gets the hypotenuse of a right triangle wrong. Multiple medieval commentators responded by concluding that the Pythagorean theorem must be incorrect.

    And if you want a really disturbing set of examples just read the end of Tamid where there is a series of disputes about the nature of the universe and the earth. In each case one group of chachamim bring an argument that is based on empirical evidence and the other bring an argument based on a proof-text from Tanach. In each case the Rabbis who bring the proof-text win. In some of the cases there we know that the group using empirical observation was simply correct.

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  10. Cool! I hadn't heard about those gemaras (once I left yeshiva I never looked at a gemara again). Next time I'm in a shul I'll look at the end of Tamid, sounds like a good laugh.

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  11. A cool mnemonic for remembering the first few digit of Pi:

    Pie! I wish I could calculate pi.

    Count the letters in each word:

    3.141592

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  12. Cool, I hadn't heard that one. I memorized the first hundred digits of pi in high school (can't think of anything more useless).

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  13. I'm also curious about your statement that you didn't learn basic geometry when you were in Yeshiva. Did you really have no math curriculum? That seems almost like child abuse.

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  14. We had a math curriculum, but it was pretty much worthless unless you had the initiative to learn the material on your own. The yeshiva did not have enough money to be able to keep decent teachers, so we had around 3 or 4 teachers in a given subject each year, some of whom were just volunteers from the community who had no experience teaching math. There was pretty much no continuity, and I remember some years where we would only get through a couple of chapters in the textbook in an entire year. The administration didn't seem to care much, as they thought that a secular education was not really important. Luckily, I had an interest in math and was able to learn it on my own, but most of my classmates couldn't wouldn't know what a cosine was if it bit them in the ass. But they got high school diplomas anyway since the administration had no problem with making up grades.

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  15. Interesting, I'm familiar with a mitnagid school that became very charedi that developed similar problems as it became more religious but I don't think there issues were as extreme although they certainly had the making-up-grades tendency and large difficulty with keeping teachers (in fact I became aware of many of the problems at the school through a friend who at one point worked there as a math teacher).

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  16. Yeah it's sad. Unfortunately, there are even worse stories. When I went to yeshiva gedola (post high school) I had classmates who had attended Lubavitch schools that did not have secular curriculums at all. I remember helping my chavrusa write a flier once, since he could hardly spell in English. So I guess I'm lucky that I got any education at all. I think this is where the government needs to step in to make sure that every child gets a basic secular education.

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  17. Well, at least in the United States, most states have laws about what schools need to satisfy at least prior to 8th grade. It gets fuzzier after that, since there are a series of court decisions that give religions large leeway after that (the primary decision said that the Amish were under no obligation to send their kids to schools after 8th grade). The homeschool movement has also given larger leeway to private schools as a consequence of their general push for fewer rules. There's a slight backlash going on in some states such as California.

    Also, my impression is that in Europe the rules are much stricter.

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  18. There may be laws on the books that require an elementary school education, but they are seemingly not enforced well enough. I know people who went to Ohelei Torah in Crown Heights who can barely read English. I suppose the government doesn't want to appear to infringe upon the free practice of religion.

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  19. I suspect that at least in New York the large and politically influential charedi population helps keep the elected officials from paying too much attention to what is going on in their schools.

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  20. You're probably right. And in my experience, most of the rabbis in charge have no problem with defrauding the government. For example, there is a system set up where people going to school overseas get financial aid from the U.S. government by getting the central lubavitch yeshiva in New York to say that the student is enrolled in their yeshiva, and the yeshiva takes a cut of the profit in exchange.

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  21. Really? Is that a federal program? If so, there might be people very interested in dealing with that who aren't in state politics.

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  22. Yeah, I'm pretty sure it was federal (Pell grants and such). There was even a lady from "Lubavitch Yeshiva" who came to our overseas yeshiva and had us fill out paperwork. I don't remember all the details, but it definitely wasn't kosher, and they were pretty cavalier about it. No harm in stealing from goyim, after all. Sickening.

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  23. I found this article:

    http://www.nytimes.com/1993/10/21/nyregion/20-new-york-schools-to-lose-us-grants.html?sec=&spon=&pagewanted=all

    it's from 1993, but I went to yeshiva in 2002 and they were still doing this scam.

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  24. Apikores, good luck with your science degree.

    Maybe this will make your day.

    http://www.sciam.com/article.cfm?id=was-einstein-wrong-about-relativity

    It's not easy reading though.

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  25. Thanks Baal Habos! I have a subscription to SciAm, so I already read that article, but thanks anyway.

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