## Number Spiral

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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
Message 1 of 3 , Mar 2, 2003
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.
• 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, 2003
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.

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@...>
Sent: Sunday, March 02, 2003 11:51 PM

> Re: the numbers generated using the number spiral strategy, at
> http://www.numberspiral.com ...the terms in each series having the
> 2,
• 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, 2003
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.
>
> 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...>
> Sent: Sunday, March 02, 2003 11:51 PM
>
>
> > 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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