Cancer treatment could become safer, and Wi-Fi networks more secure, now that a geometric puzzle has been solved.
Mathematicians have long studied the packing problem, which involves finding the best way to cram a set of shapes inside a larger object. A related puzzle is the covering problem, in which the smaller shapes completely cover the larger. An example of each might be “how many tennis balls can you fit in a box?” and “how many square tiles will cover a circular floor?”, respectively.
Now Carolyn Phillips, a physicist at the University of Michigan in Ann Arbor, and her colleagues have studied what they call the “filling” problem. Unlike the packing problem, the smaller shapes can overlap and have different sizes, but they can’t extend beyond the object’s boundary, which is allowed in the covering problem. It is essentially a mathematical version of colouring in a shape using differently sized blobs of paint.
Phillips started investigating the problem when trying to model the behaviour of a type of nanoparticle. “Physicists are used to thinking about simple hard objects that can’t overlap, while problems of covering have been traditionally more the domain of mathematicians,” she says. “Now there are nanoscale structures that can be modelled as overlapping spheres.”
Phillips and her colleagues have now explained how to solve the problem in two dimensions – for example, filling a triangle with a variety of circles. The team begin by finding the target object’s “topological skeleton”, a representation of the object to be filled as a series of lines running through its centre. The topological skeleton of a triangle comprises three lines – one beginning at each of the triangle’s points – that meet at a triple branching point. More complicated shapes can have several of these branching points.
The team place the centre of the first circle at one of the branching points in this skeleton, filling as large an area as possible. Subsequent smaller circles then sit at other points along the skeleton until the entire shape is filled.
The team is now looking for a solution to the problem in three or more dimensions. “The skeleton of a 3D shape is more complicated,” says Phillips. “Conceptually the solutions are similar, but designing an algorithm that can actually find a good optimal solution is considerably more challenging.”
Peter Cameron, a mathematician at Queen Mary, University of London, says that the new filling problem – and its solution – are interesting. However, he questions whether it will be as useful as packing and covering, which are applicable to more abstract mathematical concepts than circles or other simple shapes.
Working out how to fill a space using the minimum number of circles could have real-world applications, though. Phillips says the technique could help radiologists use the minimum number of variously sized radiation beams to treat all areas of a tumour. It could also help in the design of office Wi-Fi networks, which would use the smallest possible number of antennas and ensure that the signal does not extend beyond the building’s walls where it would be vulnerable to snoopers.
Journal reference: Physical Review Letters, DOI: 10.1103/PhysRevLett.108.198304
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