Pi in Pascal's Triangle (2014)

68 points by senfiaj 1 month ago | 14 comments
  • JohnKemeny 1 month ago
    π - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••

    That's beautiful. I wonder why the -2 is there, though. To fix it, we would need

    π = -x + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••

    where x = 2, and so it would be

    π = -1/½ + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••

    which makes the -1st triangle number ½, I guess.

    • dhosek 1 month ago
      0th triangle number and I think that can probably be connected to 1+1-1+1-1+⋯=½ somehow.
    • raldi 1 month ago
      Even after reading this, I don't understand what pi has to do with 4, 20, 56, 120, 220, 364
      • kevmo314 1 month ago
        It's the equation in the diagram:

        π = 3 + 2/3 (1/4 − 1/20 + 1/56 - ...)

      • lixtra 1 month ago
        • nimonian 1 month ago
          There's a typo in the final line of math. I think 1/7C2 should be positive rather than negative.
          • zkmon 1 month ago
            I wonder if the triangle hides any secrets related to prime numbers as well.
            • dhosek 1 month ago
              It’s possible, I suppose.

              One of my favorite proofs that the sums of each row are powers of two comes from the fact that the numbers in row n+1 are the coefficients of the powers of (a+b)ⁿ, so setting a=b=1 you get 2ⁿ (most discrete math students seeking to prove this end up reaching for induction which is a heavier proof than this).

              • nimonian 1 month ago
                I like the argument that every number in the row below is formed by summing two numbers from above. So each number above appears twice below. Hence the sum doubles.
                • hollerith 1 month ago
                  You mean, every number in the upper row contributes twice to the lower row.
                  • dhosek 1 month ago
                    Oh, that’s really nice.
                  • madcaptenor 1 month ago
                    That requires you to prove the binomial theorem first, though, and won't that need induction?
                    • dhosek 1 month ago
                      Depends on your starting point.
                  • adornKey 1 month ago
                  • 1 month ago