Professor John Horton Conway: sphere packing problem

Thursday 7th May 2020

The famous British mathematician and Princeton Professor John Horton Conway passed away on Saturday 11 April 2020 from the Coronavirus. One of his many contributions to maths was in the study of sphere packing, where mathematicians study the different ways to arrange non-overlapping spheres in a confined space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible such as packing equally sized oranges into a box with a defined volume.

In two dimensions this is called circle packing, where the aim is to fit as many equal sized circles into a confined space as possible without the circles overlapping. This is particularly relevant today since the social distancing recommendations say that people must stay a distance of at least two metres apart from each other. One way to visualise this application of circle packing is to consider each person to be at the centre of a circle of radius one metre, where the circles are not allowed to overlap. In this scenario, four circles (or people) can be ‘packed’ into a square of side length four metres.

But what about five circles? What is the smallest square which can hold five circles of radius one metre if the circles are not allowed to overlap? Hint: the arrangement must be as shown.

$$2+2\sqrt{2}$$