The Kiss Precise by Frederick Soddy
For pairs of lips to kiss maybe
Involves no trigonometry.
This 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 form 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.

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.

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.

On August 15, 1936, only a few months after Soddy's poem had been published in Nature, Thorold 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 extension of the poem to the higher dimensions

http://www.pballew.net/soddy.html