# Packed oranges, bridges and misunderstandings

David Castelvecchi, at the Scientific American blog network, wrote about a Comment article that appeared in the July 13 issue of the journal Nature.  The author, Peter Rowlett, takes note of what could happen when the mathematician “pushes ideas far into the abstract, well beyond where others would stop.”  He does this with a collection of short pieces written by various authors about the unanticipated practical value of once purely theoretical considerations.  I raise this point often in my own classes because I don’t believe these things are just happy accidents.

I agree with Castelvecchi that the piece seems to be motivated by funding issues, but it contains some nice stories.  One of the more surprising references was the one about stacking oranges.  Kepler (in 1611) conjectured that the way grocers stack oranges is the most efficient way to pack spheres. Thomas Hales finally proved this to be the case in 1998. It might seem that there is little value to the proof since we apparently already knew what we were proving.  But proofs help build systems of thought about things that we don’t immediately understand.  Packing problems are related to another kind of problem, one investigated by Newton in the seventeenth century, specifically:  how many spheres can touch a given sphere with no overlaps?  While Newton thought that in three dimensions the answer is 12, the proof of this was only given in 1953.  It took until 2003 to find that the answer to this problem in four dimensions is 24. But the answer in five dimensions can still only be specified to be between 40 and 44.  The eight dimensional version, however, was solved in 1979 when it was shown to be 240.  This result is related again to the packing of spheres (in eight dimensions) now known as the E8 lattice.  And it was the E8 lattice that was used in the 1970s for the development of a modem with 8-dimensional signals, opening up some new paths for internet signals to take.

One of the other examples in the Nature piece is one that is often on my mind, namely, topology.  In 1735, Euler began thinking topologically when he demonstrated that it was not possible to find a walk through the city of Königsberg that would cross each of its bridges only once.  Euler recognized that the only information needed to answer the question was the number of bridges and a list of their endpoints as represented in a schematic. The actual shape of the geography was irrelevant.  It is the irrelevance of distance and shape in topology that I find most provocative.  Rowlett gives a nice list of emergent applications:

Biologists learn knot theory to understand DNA. Computer scientists are using braids — intertwined strands of material running in the same direction — to build quantum computers, while colleagues down the corridor use the same theory to get robots moving. Engineers use one-sided Möbius strips to make more efficient conveyer belts. Doctors depend on homology theory to do brain scans, and cosmologists use it to understand how galaxies form. Mobile-phone companies use topology to identify the holes in network coverage; the phones themselves use topology to analyze the photos they take.

But what I find interesting about Castelvecchi’s blog is that he says this:

As much as I enjoyed the article, it must be said that picking some of the successful examples does not satisfactorily answer the broader question of whether the bulk of mathematical research is a “waste of time,” in the sense that it will never find applications anywhere. It is a legitimate question, and one that I am not qualified to answer.

I don’t think it is a legitimate question.  I think it develops out of a shared misunderstanding about mathematics in particular, and about human nature in general.  We assume to know too much about what we are doing or why.  I don’t think mathematics is driven by the same kind of problem solving as the sciences. I think it often happens within the human organism’s drive to organize. Perhaps it is grounded in the nervous system’s need to process ongoing, overwhelming and complex sensory data, which is the first way that we know anything. And I believe that this is the very reason that the riddle of a physical problem might suddenly be solved by some independently constructed mathematical system.  Mathematics is likely exploring organizational possibilities, and in this way extending the reach of cognitive processes.  More importantly, there is no way to evaluate them if they are not associated with an application, and no way to anticipate how they might be.  The idea that we are in a position to judge this fluid creativity (which is likely informed by things that are outside of our awareness) is foolhardy.  It is like believing that one could direct the evolution of human culture.