was taken from
the April 21, 2001 issue of Science News:
What are the radii
and centers of the circles marked 'a' and 'b'?
(scroll
down for more!)
It turns out that the radii are
always reciprocals of integers (here
a = 1/3 and b = 1/6),
and even more surprising (to me) is that the centers are always at rational
coords
with the same denominator (maybe lower when reduced; here (0, 2/3) and (-3/6,
4/6)).
Way back in 1638,
Descartes developed the elegant formula for
four mutually
tangent "kissing" circles with curvatures k, m, n,
p:
k^2 + m^2 + n^2
+ p^2 = (1/2)(k + m + n + p)^2 .
(There are similar
formulas to help find the centers of these circles.)
Frederick Soddy
put it into verse
293 years later:
(sent in by ProblemOfTheWeek
contestant Rick Montgomery)
Four circles to the
kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
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.
I had Mathematica draw a series of pictures and the
first
frame looks like
this; click the picture
for lots more detail!
( Here the whole
numbers represent the curvatures. )
(scroll
down for more!)
I also have a
trippy zooming movie; here are a few selected frames for you!
%;-} Dan
For more on the subject,
search at www.google.com for 'circle packing' or 'Lagarius' or 'Appollonian
tiling'.
(scroll
down for more!)
(scroll
down for more!)
