During the surprisingly seasonal cold snap at the start of 2013, which brought the country to its usual grinding halt, the inestimable Dr Grime posed a question on twitter as to how one should form the most perfectly spherical ball for a snowman and what track that would leave in the snow. As the conversation evolved several possibilities emerged, most particularly spirals, random walks and fractal paths.
The following afternoon I posed the problem to my able Year Seven group and they came up with a similar, if less mathematically expressed, set of options. We also agreed that we needed a control group and, suitably togged up as the temperature was still several degrees below freezing despite the beautiful sunshine, we headed out to investigate.
The four groups hijacked some blocks from a collapsed snowman, as one would with seed crystals, and set off to create their ‘spheres’ and paths. The snow was particularly powdery and it took longer than expected for snowballs to build in size. However, despite the lure of a nice warm classroom, the pupils insisted on trying for longer!
Group one, the control group, trudged purposefully up and down the field creating an interesting trace of footprints either side of the snowball impressions. We had speculated that their path should create a barrel shaped snowball, rather than the theoretical cylinder, and that was what they found.
In contrast, group two wandered aimlessly up the field deliberately not going in any particular direction. Random snatches of song could be heard drifting on the wind as they sang their way though most of the the top forty (I say sang, but that assumes that any given ‘hit’ has a tune to sing which is not always the case these days!). Although not a mathematically created random walk, nonetheless their path was clearly irregular and the ball was surprisingly round, especially given its asymmetric starting shape, so this was deemed to be a success.
The path created by group three, the spiral team, described a tight Archimedian spiral. You can see that their footprints in following the ball have (largely) remained inside the path. They found however that the ball gradually pivoted so that its axis remained in the direction of the spiral rather than rotating forwards. They suggested that this needed further investigation or perhaps a larger spiral.
The final group had probably the trickiest task of following a Hilbert curve. It would be fair to say that the difficulty of the task was not aided by a schism in the group which slowed the development of the ball. However, once it was agreed that we should continue despite the powdery snow, they made better progress. The rectilinear nature of their path was very evident and the ball had developed a good sphericality again despite the irregular starting shape.
It is fair to conclude, perhaps as expected, that the pupils demonstrated the three deviating paths created better spheres than the straight track of the control. However, the random walk and fractal path were distinctly better than the spiral. Perhaps they might find time for extra maths this weekend after further snowfalls; got to be better than doing history coursework!