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Number Spiral

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  • qntmpkt <qntmpkt@yahoo.com>
    Re: the numbers generated using the number spiral strategy, at http://www.numberspiral.com ...the terms in each series having the 2,2,2,2...2nd difference
    Message 1 of 3 , Mar 2 8:51 PM
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      Re: the numbers generated using the number spiral strategy, at
      http://www.numberspiral.com ...the terms in each series having the
      2,2,2,2...2nd difference row, for example one such sequence being.
      5, 11, 19, 29, etc; such terms are "curvature numbers" i.e.
      inverses of radii in Descartes circle equation. Descartes said that
      given 4 mutually tangent circles with curvatures a,b,c,d,, then (a^2
      + b^2 + c^2 + d^2) = 1/2 (a + b + c + d)^2.
      However, there are some unanswered questions regarding the means of
      generating the curvature numbers , while screening out non-curvature
      numbers. It appears that the "Number Spiral" is the appropriate
      format for further investigation in resolving the question with
      (perhaps) a plot grouping together 3 of the 4 tangential circles, but
      not necessarily the innermost circle. Anyway, here's the formula to
      find the radius of the innermost circle, given the radii A, B, C. :
      D, radius of innermost circle = C [ 2( sqrt(C/A + C/B + C^2/(AB)) -
      ))/ 2C/A + 2C/B - 1 - (C/A - C/B)^2))]. A Science News issue
      featured some of the tangential circles on the cover, ; and there are
      a number of excellent websites devoted to this subject, which can be
      accessed by plugging "Apollonian packing circles" into Google.
      Sincerely, qntmpkt.
    • Rob Sacks
      Hi qntmpkt, Thanks very much for drawing these Cartesian cuvature numbers to our attention. I find this extremely interesting. As you suggested, I found the
      Message 2 of 3 , Mar 3 2:26 PM
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        Hi qntmpkt,

        Thanks very much for drawing these Cartesian cuvature
        numbers to our attention. I find this extremely interesting.
        As you suggested, I found the Science News article in Google.
        The link is here:

        http://www.sciencenews.org/20010421/bob18.asp

        > It appears that the "Number Spiral" is the appropriate
        > format for further investigation in resolving the question with
        > (perhaps) a plot grouping together 3 of the 4 tangential circles, but
        > not necessarily the innermost circle.

        Can you describe in a little more detail how you would make
        a plot on the number spiral that corresponds to the packed
        circles?

        Best regards,

        Rob

        ----- Original Message -----
        From: <qntmpkt@...>
        To: <primenumbers@yahoogroups.com>
        Sent: Sunday, March 02, 2003 11:51 PM
        Subject: [PrimeNumbers] Number Spiral


        > Re: the numbers generated using the number spiral strategy, at
        > http://www.numberspiral.com ...the terms in each series having the
        > 2,
      • Dick
        Hello, The relationship between the Spiral and the Apollonian Circles is amazing and intimate! I ve written something up on it here
        Message 3 of 3 , Mar 4 8:03 PM
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          Hello,

          The relationship between the Spiral and the Apollonian Circles
          is amazing and intimate!

          I've written something up on it here

          http://www.imathination.net/APPVeins.htm

          Regards,

          -Dick


          --- In primenumbers@yahoogroups.com, "Rob Sacks" <editor@r...> wrote:
          > Hi qntmpkt,
          >
          > Thanks very much for drawing these Cartesian cuvature
          > numbers to our attention. I find this extremely interesting.
          > As you suggested, I found the Science News article in Google.
          > The link is here:
          >
          > http://www.sciencenews.org/20010421/bob18.asp
          >
          > > It appears that the "Number Spiral" is the appropriate
          > > format for further investigation in resolving the question with
          > > (perhaps) a plot grouping together 3 of the 4 tangential circles,
          but
          > > not necessarily the innermost circle.
          >
          > Can you describe in a little more detail how you would make
          > a plot on the number spiral that corresponds to the packed
          > circles?
          >
          > Best regards,
          >
          > Rob
          >
          > ----- Original Message -----
          > From: <qntmpkt@y...>
          > To: <primenumbers@yahoogroups.com>
          > Sent: Sunday, March 02, 2003 11:51 PM
          > Subject: [PrimeNumbers] Number Spiral
          >
          >
          > > Re: the numbers generated using the number spiral strategy, at
          > > http://www.numberspiral.com ...the terms in each series having
          the
          > > 2,
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