Imagine a circle; inscribe within it three other circles congruent to each other and of maximum radius; proceed similarly within each of these circles and within each interval between them, and imagine that the process continues to infinity.

Apollonian packing of circles

Passato sotto le mani di Euclide, Cartesio e del meno noto ma molto più recente matematico polacco Sierpiński, identificato come “frattale” solo dopo che Mandelbrot (anche lui polacco) ha coniato il termine, l’impacchettamento ha avuto una soluzione sotto forma di versi da parte del prof. Soddy con una pubblicazione su Nature:

Four 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.

Apollonian packing of spheres

Four circles to the kissing come,
The smaller are the bender,
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb.
There’s now no need for the rule of thumb.
Since zero bends a 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.

Nella procedura immaginata da Apollonio la taglia dei cerchi inseriti (procedura ovviamente generalizzabile in dimensione n, come da terza strofa della poesia) tende a zero e, va da sé, l’insieme di questi oggetti tende ad occupare perfettamente lo spazio disponibile. Ebbene, a chi dovesse trovare questo inseguimento dell’impacchettamento perfetto solo un gioco matematico, sappia che invece ci sono concrete possibilità di utilizzo