- 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 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

Can you describe in a little more detail how you would make

> 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.

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, - 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,