WOBBLING tables are one of life’s little irritations, like odd socks and toast that falls butter side down. But not for theoretical physicist André Martin. While most of us would be content to prop up a shaky table with a folded napkin or beer mat, Martin has found a far more elegant solution. “I will never use a piece of paper again,” he says.
Martin, who recently retired from the CERN particle physics laboratory near Geneva, Switzerland, has used mathematics to prove what every wine-stained victim of a wobbly table always wished they had known. To set your table with four legs firmly on the ground, all you need to do is turn it. Even on uneven ground, dining nirvana is never more than 90 degrees away.
CERN may seem an unusual place to study such a down-to-earth problem. Deep underground robots are helping physicists align the hundreds of massive magnets that will soon be part of the world’s most powerful particle accelerator. Above ground, scientists taking a break face a different sort of balancing problem. The terrace outside CERN’s main cafeteria is so uneven that the tables constantly wobble. Over the years, many a glass of fine wine has spilled on physicists’ laps.
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Fed up with leaning on tables only for his coffee to splash, Martin decided to tackle the problem. He first discovered that it was possible to stop a table wobbling simply by turning it in the mid-1990s. In fact it worked so well that he knew there must be an explanation and Martin soon found one based on geometry – at least for the case of a perfectly square table. Still, as a mathematically-minded theorist, he wanted a proper proof. “I realised I wasn’t being rigorous enough,” he says.
In a paper due for publication in Physics Letters A, Martin sets out his proof in eight pages of dense trigonometry. While working on it, he discovered that his idea had been around in various forms for several decades, though no one had published a rigorous proof. In fact it turns out that mathematicians have produced an extensive unpublished corpus of related results, including – it almost goes without saying – techniques for stabilising tables in a space with four dimensions or more.
So how does it work? Martin first assumes that the table legs are all the same length, so the wobble is caused by uneven ground. He also supposes that only one table leg is in the air. The secret, he discovered, is to turn the table while keeping three of its legs continuously on the ground. He proved that at some point during this dragging, the fourth leg will inevitably end up touching the ground too (see Diagram). Once all four legs are on the ground, voila, the table will stop wobbling.
Practice makes perfect
Ever since formulating his proof, Martin has found his outdoor meals more enjoyable. “I tested it for years at the CERN cafeteria,” he says. “It works.”
Of course, not all table legs sit in a square. Fortunately a team of mathematicians in Australia and New Zealand has come up with a slightly different technique for stabilising tables whose legs form a rectangle. For their thought experiment, Burkard Polster of Monash University in Melbourne and his colleagues also assume that the wobble is caused by uneven ground rather than legs of different lengths. They imagine dragging the table while keeping two diagonally opposite feet on the ground and tipping it so that the other two feet are at equal heights over the ground at all times. They too have shown that it is always possible to find a stable spot with four legs on the ground.
One drawback is that their technique is hard to apply – you have to constantly evaluate the heights of the two hovering feet and keep them equal to each other – but it has the advantage of working reliably with any rectangular table, rather than just with perfectly square ones. Their proof, which will appear in The Mathematical Intelligencer magazine, also shows that it is possible to stabilise a table while turning it on the spot – handy for helping you deal with tables anchored by those umbrella poles that stick out the middle.
As you might expect from theoretical physicists and mathematicians, putting their proofs into practice is no picnic. For a start, they don’t tell you how far to turn your table to stop it wobbling – only that it is possible to find a stable position. The other catch is that, with both techniques, the table’s final position may be wobble-free, but still on a slope. “I am not proving that the table will be horizontal,” says Martin.
Still, Martin’s technique was a success when I tested it on the sidewalk tables outside several cafes in Washington DC. The only obstacle was that the tables in some cafes were crammed tightly together, with little room for turning. Incidentally, the method works with wobbly chairs too, though you might end up having to sit sideways.
So what’s next? Will mathematicians address more of the frustrating little geometric discrepancies in our daily lives? Might new theorems help us assemble flat-pack furniture? Could new lines of research shed light on why packing your suitcase always seems so much harder at the end of a trip than it was at home?
For now, both Martin and Polster’s teams are planning to extend their studies to non-rectangular tables, such as trapezium-shaped tables common in school classrooms, and to tables with legs of slightly different lengths. In the latter case, Polster points out that “there is not a chance in hell it will work out if the ground is flat.” No amount of dragging will stop a table with uneven legs wobbling on flat ground. But sometimes, irregular surfaces and uneven legs go well together, and Polster and his team would like to understand exactly what conditions make that possible.
If nothing else works, the good old folded napkin might still be your best bet.
