|The Kiss Precise by Frederick Soddy
In Nature #137, 1936
For pairs of lips to kiss maybe
Four circles to the kissing come.
Expanded to spheres : (Nature #139, 1937)
To spy out spherical affairs
2(k1² + k2² + k3² + k4²) = (k1 + k2 + k3 + k4)²
When one circle is around the others, its curvature is taken < 0.
When one circle is a straight line, its curvature is 0.
This formula was first established by Descartes, but brought to fame by Soddy with a poem "The Kiss Precise", and extended to spheres, and n dimensions.
It gives the radius of the 4th circle tangent to three given tangent circles :
r4 = r1r2r3 / ( r1r2 + r1r3 + r2r3 ± 2√(r1r2r3(r1 + r2 + r3)) )
We'll give here the proof by Philip Beecroft (1842).
First of all, consider two sets of 4 circles mutually tangent :
Circles (C1) to (C4) with radii r1 to r4 and curvatures k1 to k4
Circles (Γ1) to (Γ4) with radii R1 to R4 and curvatures m1 to m4
(Γ4) is the incircle of triangle ABC with sides a, b, c.
Considering in the same way circles (Γ1) to (Γ3)
with centers O1 to O3 and incircle C4 in O1O2O2 :
k4² = m1m2 + m2m3 + m1m3
As well as all formulas from swapping the indexes (considering excircles).
We get then the sum of the squares of ki's and mi's :
∑ki² = 2∑mimj and ∑mi² = 2 ∑kikj
Calculate then the square of the sum of ki's :
(∑ki)² = ∑ki² + 2∑kikj = ∑ki² + ∑mi²
And similarily (∑mi)² = ∑mi² + ∑ki²
Hence (∑ki)² = (∑mi)², and as they are >0 : ∑ki = ∑mi
Calculate then the value :
(k1 + k2 + k3 + k4)(k1 + k2 + k3 - k4) = (k1 + k2 + k3)² - k4² = k1² + k2² + k3² + 2m4² - k4² =
∑2,3,4mimj + ∑1,3,4mimj + ∑1,2,4mimj - ∑1,2,3mimj + 2m4² = 2(m1m4 + m2m4 + m3m4) + 2m4² = 2m4(∑mi)
And, as ∑mi = ∑ki :
k1 + k2 + k3 - k4 = 2m4
Squaring the 4 similar relations and summing up, all terms as kikj cancel and remains :
∑ki² = ∑mi²
Hence (∑ki)² = ∑ki² + 2∑kikj = ∑ki² + ∑mi² = 2∑ki²
That is finally the Descartes relation :
(∑ki)² = 2∑ki²