Title: Packing circles and spheres

"Konjam adjust pannikkunga". "Thoda adjust kar lo bhai". "Solpa adjust maadi".
These are very familiar sentences to anyone who travels on buses and trains in India. The suburban trains in Mumbai demand great expertise on the part of their regular commuters in this regard.  When it comes to packing a great number of people in a small volume, Indian metros perform what seem to be daily miracles.  
The mind of any mathematician seeing these exercises must be racing, calculating the packing density, and checking whether our trains achieve the limit of how much packing is possible.  
Can we really calculate such a thing ? But people are not very regularly shaped. If we were to consider a coach as a cube, and people as built of small cubes, it is easy to calculate the packing density. However, people are more like cylinders, and packing cylinders inside a cube must waste some space, surely.
Indeed, we may wonder how curved shapes could be fitted inside angular ones. Typically mathematicians approach such questions by first considering regular shapes. The simplest to consider in this regard would be spheres inside cubes. How many unit spheres can be packed inside a cube of side (say) 4 ? (For convenience, we assume that these are hard incompressible spheres, to be accommodated wholly.) And having packed them in, what is the largest sphere that can be packed into the remaining space ?
Once again, it is simpler to answer such questions in two dimensions before we attmpt them in three dimensions. Reformulating the question, we can ask: how many unit circles can we pack inside a square of side (say) 4 ? And what is the radius of the largest circle that can be squeezed in to the remaining space ?  
It is easy to see that 4 unit circles can be accommodated inside a square of side 4, with each circle touching two of its neighbours.  (See Fig 1). Now note that there is just about enough room for another little circle at the centre of the square. What is the radius of the inner circle in Fig 2 ?  
Consider the diagonal of the lower left square, as in Figure 2.  The segment AO from the centre of the circle A to O, the centre of the Figure, is \sqrt{2}, since it is the hypotenuse of the right angled triangle formed by AB and BO. (AB is 1, being the radius of the unit circle, and BO is 1 since it is a quarter of the Figure, a square of side 4.) Subtracting the radius of the unit circle, we find that the radius of the inner circle is \sqrt{2} - 1.  
We can now work out the three dimensional case, and we find that 8 unit spheres will pack into the corners of a cube of side 4, and the largest sphere that will fit into the centre has a radius \sqrt{3} - 1. (See Figure 3).  
In general, in the corners of an n-cube of side 4, we can pack 2^n unit n-spheres. What about another sphere at the centre, of radius \sqrt{n} - 1 ? The answer is yes, for n < 9. The surprising fact that this does not happen in 9-space was pointed out by Leo Moser.  
Once we start thinking of circles and spheres touching each other, is it difficult to stop. Above, we were looking for unit circles confined inside a square. One thing is to liberate them from inside the square, and ask: 
What is the largest number of unit circles that can touch a unit circle ?
The answer is 6, see Figure 4. In three dimensions, the answer is 12, where the 12 spheres are arranged around a 13th sphere, their centres at the corners of an imaginary icosahedron (Fig 5). Apparently Isaac Newton knew this (according to a reported conversation), but a formal proof had to wait until 1874.  
The next step is to move away from unit circles:
How many circles can be placed (on the plane) so that
(a) each circle touches all the others, and
(b) each pair touches at a different point ?
There are only two ways of doing this: either three circles surround a smaller one (Fig 6a), or three circles are inside a larger one (Fig b). There is a remarkably elegant formula showing the relationship between these 4 circles. The curvature of a circle is radius r is 1/r. Let a, b, c, d be the curvatures of the circles above. Then 
(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2).
For the curious, in three dimensions we have 5 mutually touching unit spheres, and the formula relates the curvatures in a very similar way: (a+b+c+d+e)^2 = 3(a^2+b^2+c^2+d^2+e^2).
Apparently Descartes knew of this relationship, but it was Frederick Soddy who first published this result in 1936. Interestingly, Soddy, who got his Nobel prize in 1921 for his discovery of isotopes, wrote it in the form of a poem !