The Kiss Precise, Soddy's Circle Theorem

 Soddy's formula is another example of Stigler's law of eponymy, "No scientific discovery is named after its original discoverer." Stigler named the sociologist Robert K. Merton as the discoverer of "Stigler's law", so as to avoid this law about laws disobeying its very own decree.

Soddy's formula is about the relationship of the radii of four mutually tangent circles. The formula is sometimes called the "Kissing Circles Theorem". If four circles are all tangent to each other, then they must intersect at six distinct points. The first demonstration of this relationship between four mutually tangent circles (actually, one can be a line) was in 1643. Rene Descartes sent a letter to Princess Elisabeth of Bohemia in which he showed that the four radii, r₁, r₂, r₃, r₄, must be such that \( \frac{1}{r^2_{1}}+ \frac{1}{r^2_{2}}+\frac{1}{r^2_3}+\frac{1}{r^2_4}= \frac{1}{2}(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4} )^2\)

For this reason the theorem is often called Descarte's circle theorem. The figure shows four circles all externally tangent to each other, but could also be drawn with three tangent circles all inside, and tangent to, a fourth circle. The bend of this externally tangent circle is given a negative value, and thus the same equation provides its radius also.

The equation can be written much more easily, and usually is, using a notation of "bend". For each value let the "bend" equal the reciprocal of the radius, then \(\frac{1}{r_1} =b_1\)With this notation the formula can be written as \(b^2_1 + b^2_2+b^2_3+b^2_4=\frac{1}{2}(b_1+b_2+b_3+b_4)^2\).

It seems that it may also have been discovered about the same time in Japan. In the book, Sacred Mathematics: Japanese Temple Geometry, by Fukagawa Hidetoshi and Tony Rothman, there is an illustration of a complicated pattern of nested congruent circles, for which knowledge of the theorem would seem to be required, on a wooden tablet. It was a practice during the Edo period in Japan that people from every segment of society would inscribed geometry solutions on wooden tablets called sangaku and hang them as offerings in temples and shrines.
The Theorem was rediscovered and published in the 1841 The Lady's and Gentleman's diary by an amateur English Mathematician named Phillip Beecroft. Beecroft also observed that there exist four other circles that would each be mutually tangent at the same four points. These circles would have tangents perpendicular to the original circles tangents at each point of intersection. Both sets of Beecroft's circles are shown in this illustration from Mad Math.
Beecroft's circles are related to the use of a geometrical inversion in a circle which will invert the inner tangent circle to become an outer tangent circle. The circle of inversion between the two is the circle Beecroft uses that passes through the three points of tangent in the other three circles. (A nice explanation and illustration of this is at this AMS site.

In 1936 Sir Fredrick Soddy rediscovered the theorem again. Soddy may also be known to students of Science for receiving the Nobel Prize for Chemistry in 1921 for the discovery of the decay sequences of radioactive isotopes.  According to Oliver Sacks' wonderful book, Uncle Tungsten, Soddy also created the term "isotope" and was the first to use the term  "chain reaction".  In a strange "chain reaction" of ideas, Soddy played a part in the US developing an atomic bomb.  Soddy's book, The  Interpretation of Radium, inspired  H G Wells to write The World Set Free in 1914, and he dedicated the novel to Soddy's book. Twenty years later, Wells' book set Leo Szilard to thinking about the possibility of Chain reactions, and how they might be used to create a bomb, leading to his getting a British patent on the idea in 1936.  A few years later Szilard encouraged his friend, Albert Einstein, to write a letter to President Roosevelt about the potential for an atomic bomb.  The prize-winning science-fiction writer, Frederik Pohl, talks about Szilard's epiphany in Chasing Science (pg 25),

".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb.  There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row.  Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head."

Perhaps Soddy's name is appropriate for the formula if only for the unique way he presented his discovery. He presented it in the form of a poem which is presented below.

The Kiss Precise
by
Frederick Soddy

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 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.

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.

In _Nature_, June 20, 1936

One may notice in the last verse that Soddy generalizes the theorem to five spheres. The extended theorem becomes: \[ b_1^2+b_2^2+b_3^2+b_4^2+b_5^2 = \frac13(b_1+b_2+b_3+b_4+b_5)^2. \]

Later   another verse was written by Thorold Gosset to describe the even more general   case in N dimensions for N+2 hyperspheres of the Nth dimension.

On August 15, 1936, only a few months after Soddy's poem had been published in Nature,  Gosset sent a copy of the poem to Donal Coxeter on the occasion of his wedding in the Round Church in Cambridge.  Gossett enclosed in his wedding congratulations, and his extension of the poem to the higher dimensions which were Coxeter's special area of study. It would be published in Nature the following year,
 

The Kiss Precise (Generalized) by Thorold Gosset

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.

In _Nature_
    January 9, 1937.

Fred Lunnon sent me a kind note correcting a typing oversight, and adding that

"The original result generalizes nicely to curved n-space with
curvature v [e.g. v^2 = +1 for elliptic space, -1 for hyperbolic]
in the form



        \((\sum_i x_i)^2 - n \sum_i x_i^2 = 2n v^2\)




where \(x_i\) denote the curvatures of n+2 mutually tangent spheres.
Example: n = 2, v = 0, x = [-1,2,2,3] is one solution, corresponding to
a unit circle in the plane enclosing circles of radii 1/2,1/2,1/3.



See Ivars Petersen "Circle Game" in Science News (2001) \bf 159 (16) p.254"

 

Fred admits he wasn't the first to prove this, but did manage to replicate it on his own (which impresses the heck out of me)... but THEN....... wait for it.... He wrote another poem verse to accompany this extension to higher dimensions...

  The Kiss Precise (Further Generalized) by Fred Lunnon



    How frightfully pedestrian

    My predecessors were

    To pose in space Euclidean

    Each fraternising sphere!

    Let Gauss' k squared be positive

    When space becomes elliptic,

    And conversely turn negative

    For spaces hyperbolic:

    Squared sum of bends is sum times n

    Of twice k squared plus squares of bends.

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