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Science

Pi In The 4th Dimension 38

Posted by chrisd from the almost-isn't-equal-but-interesting dept.
Anonymous Coward writes "Hoy! There is some crazy guy who is all set to prove that the value of the mathematical constant Pi is same for all dimensions. He has calculated Pi for the 4th dimension - and it..it's almost equal - 3.1447817532635 wheras the calculated value of Pi in 2 dimensions (circle) is 3.1416999189943. Math nerds can calculate the percentage error. The project is on SourceForge.net and they even have an online version "
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Pi In The 4th Dimension More

Pi In The 4th Dimension

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  • Pi (Score:3, Informative)

    by fava ( 513118 ) on Thursday July 04, 2002 @02:54PM (#3823496)
    Pi isn't different for different geometries, its just the series converge slower for the higher dimentions.

    The only way pi could be different was if space itself was not "flat" but curved.

  • Failed? (Score:2, Funny)

    by undeg chwech ( 589211 )
    So he's all set to prove they're the same. A simple web page shows they're different.

    Doesn't sound like a very exciting story to me.
  • I minored in math and I'm not sure just what is meant by Pi in the fourth dimension. By definition a circle is a two dimensional object. By definition Pi = C/r where C is the circumference of the circle, and r is the radius of the same circle. Therefore Pi by definition is the ratio of two properties of a two dimensional object.

    Which leaves me (or us) with the question of what does the fourth dimension have to do with the value of Pi?

    Just showing my ignorance...
    • Well surface area of a sphere is S=(4/3)pi(r^3)
      so that means pi=S/r^3(3/4)
      But its the same pi but it you have to modify it with a 4/3 and I'm sure you could expand it another
      dimension and you'd need a different modifier, don't feel like calculating it right now, but I don't see any reason to think the actual value of pi would ever change.

  • If you did more cycles you would see that all the pi's are about equal and closer to the real pi. It just uses a brute force method so you need more cycles to see that they are more equal.

  • No, that's not pi (Score:4, Informative)

    by GCP ( 122438 ) on Thursday July 04, 2002 @03:36PM (#3823654)
    Pi is 3.1415926...., not either of the values claimed in the article. Approximations of pi could perhaps have the values claimed in the article. I could approximate pi as 3, but that wouldn't make pi = 3.

    • Perhaps that is why the author of the article said that you could calculate the margin of error...
      • He was referring to the difference between the values of pi for different dimensions. The question he was addressing was whether pi was the same for every dimension. He offered a value for 3D, a different value for 2D, and said you could calculate the margin of error.

    • Thats because they didn't run the calculation for any real length of time. Behold:

      --[~]-- ./2d3d4d_1

      INPUT
      Calculate the Pi for circle, sphere and hypersphere.
      Let us start the verification for 2d, 3d and 4d, Give Cycles
      Cycles = 100000000000

      CIRCLE
      Area of circle by brute force method = 78539815867
      Radius = 158113.883008
      Calculated value of Pi = 3.141593

      SPHERE
      Volume of sphere by brute force method = 52359877534
      Radius = 2320.794417
      Calculated value of Pi = 3.141593

      HYPERSPHERE
      Enclosure of Hypersphere by brute force method = 30842511725
      Radius = 281.170663
      Calculated value of Pi = 3.141593

      The longer it runs for, the closer it gets. Whether or not this actually proves anything is, of course, up to debate.

  • proving identities (Score:4, Insightful)

    by Saint Nobody ( 21391 ) on Thursday July 04, 2002 @03:37PM (#3823655) Homepage Journal

    i had to go through the source to figure out what they meant by pi in higher dimensions. at first i assumed it had to be something interesting, since they bothered to post it, but it's not. they're just useing simple calculations to find the volume of spheres and hyperspheres, and plugging that into the known, proven formulas to find pi.

    for a circle, A=pi*r^2
    for a sphere, V=4/3*pi*r^3
    for a hypersphere, V=pi^2*r^4/2

    by "proving" that pi is the same in multiple dimensions, they merely demonstrate these well-established identities. of course, since they're using a clumsy "brute force" method to initially calculate the volume, they're not actually proving anything, since they only get an approximate answer.

    • Not only that, but he's numerically integrating it up I believe, so the actual answer will diverge from the true one due to rounding errors, probably more so in the higher dimensions.

      OTOH the value of 'pi' (ratio of circumference to diameter) is actually BIGGER in hyperbolic geometry, and smaller in spherical plane geometry than the normal value, and varies with the size of the circle. Atleast that's something vaguely interesting.

    • From what I read on the website I think you are correct about what they are doing. However, the idea of a "multidimensional pi" is actually kind of interesting.

      The pi we call pi is the one for 2-dimensions, say pi(2). This is curcumference/diameter.

      For 3 dimensions, pi(3), the extension that seems the most natural to me is (outer surface area of a sphere)/(area of circle of same diameter). I don't want to go dig up an analytic geometry book right now to look up the formulas, does anybody actually remember them?

      For pi(4), I think it should be something like (3D "circumference")/(volume of sphere). While the 2D "circumference" was no trouble to figure out, just the outer surface area, I am not sure what the 3D circumference would be.

      It would be really neat if pi = pi(2) = pi(3) = pi(4) = pi(5) = ..., but something else might be more informative about geometry in general.

    • I like the fact that your sigline is a C program to print itself ... it nicely illustrates your subjectline!

  • Math nerds can calculate the percentage error.

    Why the percentage error is exactly 3.14157429...%

  • Pi is defined to be the ratio of the circumference of a circle to its diameter. pi = c/d. Okay, we all know that.

    Using calculus, you can determine the area of a circle in terms of pi.

    Using calculus, you can determine the volume and surface area of a sphere in terms of pi.

    And so on, and so on. Therefore, why would anyone think pi would be any different for higher dimensions? Are they also speculating that calculus is a load of poppycock?

  • OOOOOAAAAHHHHH (drool) N-Dimensional Pi......
  • pi == 3

    God told me so
  • I'm impressed!
  • pi in 2d is defined in terms of a ratio of the circumference of a circle to it's diameter.

    pi in 3d is conventionally defined in terms of the ratio of the 2d circumference of a circle to it's 2d diameter. It's pi, damn it, it is a constant.

    But you can also use 4piR^2 for the surface area of a sphere, and 4/3piR^3 for the volume, which has been mathematically proven for quite some time. The authors of this program appear to be defining 3D pi wrt these ratios... physics geeks will note that the units balance out nicely.

    As for 4D pi... I can only assume the ratios are something like npiR^3 surface and mpiR^4 for volume, seeing as how the units must balance and hypergeometry wasn't an emphasis at my college.

    Still, though. Writing programs to calculate pi is a fun little project that everyone seems to do at some point in their high school or college career. I did pi in HS. In college I did 4D Tic Tac Toe. These people appear to be doing both. More power to them.

  • As far as I know pi is just a number: it exists and it has his own value, like 666, sqrt(2), e, i, etc.

    I suppose that the main reason why it has a name is that lots of geometrical constant happen to belong to Q(pi) (i.e. the minimal subfield of the complex field that contains all rational numbers and pi); since Q(pi) is also an (infinite dimensional) vector space on Q, and most of the useful constant belongs to the subspace generated by 1 and pi, we can just write them as a*1+b*pi and we're able to work with them in an easier way.

    The "definition of pi" is just like the definition of any other real number, as a Dedekin section (not sure it's the name you use in english), and it happens not to say much about its value. Then when you have real numbers you can say whether two of them are the same or not, so you can demostrate that, e.g. the ratio between circonference and diameter of a circle is the same number as the limit of a few series, but calculating an aproximation of the first or some term of the second won't help much in this demonstration.

  • What I find a bit confusing is that what is meant by "higher dimension pi" is not defined anywhere.

    The main point of the project seem to dabble with different algorithms to calculate the decimal development of pi. While this is interesting, they are in fact very low level since the current record is in the millions (billions ?) of number. There are whole books dedicated to this sole topic.

    Xavier

  • So, someone got fed up running endlessly in circles, and decided that he wanted another slice of pi. Got four dimensions, and did a new number.

    I just what type of pi. Personally, I go for chocolate cream.
  • from http://www.joyofpi.com/pifacts.htm

    Pi.
    1.The Greek letter P or p, corresponding to the roman p.

    2.A number, represented by said letter, expressing the ratio of the circumference of a perfect circle to its diameter. The value of pi has been calculated to many millions of decimal places, to no readily apparent purpose: no perfect circles or spheres exist in nature, since matter is composed of atoms and therefore lumpy, not smooth. Nature
    herself sometimes takes to rounding off the more extreme decimals of numbers when they get sufficiently small, as Prof. Heisenberg has pointed out. However, the continued extension of pi provides a harmless exercise of computer power which would otherwise be misused playing Quake or surfing pointless web sites.

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I'd rather just believe that it's done by little elves running around.

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