Inside a pipe 3 units in diameter are two pipes, of diameters 2 and 1.
geometry/tangent.circles New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Inside a pipe 3 units in diameter are two pipes, of diameters 2 and 1. What is the diameter of the biggest pipe that will also fit inside the first along with the other two? Suppose you had a cylindrical pipe labeled X, 3 units in diameter. Inside of X, you insert 2 pipes, the first pipe labeled Y is 2 units in diameter, the second pipe labeled Z is 1 unit in diameter. All pipes are infinitely thin, and both Y and Z obviously fit snugly inside X. Question, What is the diameter of the biggest pipe that will also fit into X along with Y and Z? -- David Matusow <matusow@nenya.gsfc.nasa.gov> [ Editor: the first paragraph is the editor's summary. ]
Show solution
Steve Preston (spreston@gte.com) responded with the correct answer (see
below). I cannot refrain from posting the following cool theorem which
gives the result immediately:
Given (n+2) distinct, mutually tangent (n-1)-spheres with radii 1/|si|,
then
n Sum[si^2] = Sum[si]^2,
where the si are real and their signs are chosen so that externally
tangent spheres have the same sign, internally tangent spheres have
opposite sign, and hyperplanes have si=0. (If these signs cannot be
chosen, the theorem does not apply. This happens iff there are points
at which more than two spheres are tangent, making the problem
underdetermined.)
Applied to the current problem this yields
2 (1/3^2 + 1/2^2 + 1^2 + s^2) = (-1/3 + 1/2 + 1 + s)^2.
Thus (s - 7/6)^2 = 0. So the diameter is 6/7.
(Yes, we used diameters instead of radii. Go figure.)
Nevertheless I think the answer to the problem AS STATED should be 3.
This theorem is in a book by D. Pedoe (Intro to College Geometry(?)) in
the section 'A Theorem of Descartes'.
-- Jim Ferry <jpf@stickycamelot.rockefeller.edu>
[ Editor: Steve Preston's answer is not in the collected material. ]
----
I remember reading a bit of doggerel about this: (n=2)
Four circles to the kissing come,
the smaller are the benter;
the bend is just the inverse of
the distance from the center.
Since zero bend's a dead straight line,
and concave bends have minus sign,
the sum of the squares of the four bends
is half the square of their sum.
-- David Moews <moews@math.berkeley.edu>
[ Editor: the complete poem of which a stanza is quoted in this entry,
with Gosset's extension, as posted in February 1996. ]
THE KISS PRECISE
For pairs of lips to kiss maybe
Involves no trigonometry.
`Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
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._
To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
_The square of the sum of all five bends
Is thrice the sum of their squares._
--- Frederick Soddy (1936)
And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears.
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares---
As n + 2 such osculate
Each with an n + 1 fold mate
_The square of the sum of all the bends
Is n times the sum of their squares._
--- Thorold Gosset (1936)
-- David Moews <dmoews@xraysgi.ims.uconn.edu>