From: Clive Tooth
Subject: Re: Fourth Disk
Date: Wed, 13 Oct 1999 21:38:44 +0100
Newsgroups: sci.math
Keywords: Kiss precise -- the four radii of tangent circles
Leroy Quet wrote:
> Consider three disks, A, B, and C, with radii a, b, and c. No disk
> overlaps, and each disk touches the other two. Consider a fourth disk,
> D, that does not overlap any of the other three disks, but touches
> each of them.
> What is the radius, d, of D?
> Also, consider the triangle with vertexes at the centers of A, B, and
> C. What is the name of the point, relative to the triangle, that is at
> the center of D?
> Thanks,
> Leroy Quet
"bend" = "reciprocal of radius"
If the four bends are a, b, c and d - then:
2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2
THE KISS PRECISE
Four circles to the kissing come,
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line,
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
- Frederick Soddy
--
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
End of document
==============================================================================
From: jr@redmink.demon.co.uk (John R Ramsden)
Subject: Re: Fourth Disk
Date: Sat, 16 Oct 1999 05:41:32 GMT
Newsgroups: sci.math
On Wed, 13 Oct 1999 21:38:44 +0100, Clive Tooth
wrote:
[above article --djr]
Someone in a recent sci.math thread (relating to Ramanujan?) lamented
the "Western-centric" view of math. Now Soddy's Theorem is an example
of something discovered in the West that could well have been known
centuries ago in Japan.
A Scientific American article within the last year or two was devoted
to these sort of mutual contact problems, which apparently the Japanese
devised in profusion and drew on their walls.
Cheers
---
John R Ramsden # "Off the top of my head, I'd say that
# my father was the better swasher,
(jr@redmink.demon.co.uk) # but I was the better buckler."
#
# Douglas Fairbanks Jr
==============================================================================
From: Clive Tooth
Subject: Re: Fourth Disk
Date: Fri, 15 Oct 1999 23:00:01 +0100
Newsgroups: sci.math
John R Ramsden wrote:
[above article quoted --djr]
Yes! Sangaku! I have one, of my own design, 2 meters from me at this
moment.
See
http://www2.gol.com/users/coynerhm/0598rothman.html
--
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
End of document
==============================================================================
[From Coxeter's "Introduction to Geometry", Wiley, 1980:
"Let us now consider four circles E1, E2, E3, E4, tangent to one
another at six distinct points. Each circle E_i has a _bend_ e_i,
defined as the reciporcal of its radius with a suitable sign attached,
namely, if all the contacts are external...the bends are all positive,
but if one circle surrounds the other three... the bend of this
largest circle is taken to be negative; and a line counts as a circle
of bend 0. In any case, the sum of all four bends is positive.
"In a letter of November 1643 to Princess Elisabeth of Bohemia, Rene
Descartes developed a formula relating the radii of four mutually
tangent circles. In the "bend" notation it is
2( e1^2 + e2^2 + e3^2 + e4^2 ) = (e1 + e2 + e3 + e4)^2.
"This _Descartes circle theorem_ was rediscoverd in 1842 by an English
amateur, Philip Beecroft, who observed that the four circles E_i
determine another set of four circles H_i mutually tangent at the same
six points...
[proof of theorem given, using all eight bends symmetrically]
"In 1936, this theorem was rediscovered again by Sir Frederick Soddy,
who had received a Nobel prize in 1921 for his discovery of isotopes. He
expressed the theorem in the form of a poem, _The Kiss Precise_[*], of
which the middle verse runs as [shown above]."
Footnote is:
"[*] Nature 137 (1936) p 1021; 139 (137) p. 62. In the next verse,
Soddy announced his discovery of the analogous formula for 5 spheres
in 3 dimensions. A final verse, added by Thorold Gosset (1869-1962)
deals with n+2 spheres in n dimensions; see Coxeter, Aequationes
Mathematicae, 1 (1968) pp 104-121."
--djr]