Do you always get the wobbly table at restaurants and cafés? Don't despair. A physicist has proved that, within reasonable limits, it is always possible to rotate the table to a position where all four legs stand solidly on the ground.

Wobbly tables can always be fixed - if they have four legs.© Punchstock

André Martin was moved to study the problem because he was fed up with the wobbly tables at CERN, the European particle physics laboratory in Geneva, Switzerland, where he works on abstruse problems in high-energy physics.

Anyone who drinks a cup of coffee on the terrace of the CERN cafeteria, says Martin, discovers that the tables usually have only three feet resting on the ground, so that the slightest touch spills your drink.

Time after time, Martin would find himself rotating the table to look for a stable position. "I've always been able to find one," he says. "People are sometimes amazed that it works."

More than ten years ago, Martin decided to see if he could find some proof that a stable state always exists. He believed that he'd found one, and even presented it at a summer school in 1998, but he never wrote it up and discovered that in any case it wasn't completely correct.

Now Martin believes he has a more watertight case, and this time he has gone public¹. "I had the feeling that mathematicians were interested," he explains.

Square solution

To turn the problem into one that could be described mathematically, he had to make some simplifying assumptions. For one thing, he considered only tables with legs at the corners of a square. And he assumed that each leg touches the ground only at a single point.

The ground itself, meanwhile, is more amenable than it often is in nature. It has any number of bumps, but no really sharp curves or changes in height: the inclination between any two points on the surface is never more than 15°.

"Compared with nature, it is smooth," Martin admits. But he adds that a poorly levelled concrete surface or a bumpy lawn would probably meet these conditions. So long as that is so, Martin has demonstrated that there will always be a way of making all four of the table's legs touch the ground.

In general, that won't leave the tabletop perfectly horizontal, but any slope will be small.

David Dallman, a CERN librarian, asked Martin whether his proof could be extended to tables with non-square arrangements of four legs. Martin suspects this can be done if all four feet lie on a circle, but he doesn't yet have a rigorous proof.

Whether it will help during the coffee breaks at CERN is another matter: the ground there might be too irregular. "The trouble with the terrace is that there is both grass and paving slabs," Martin says. 

CERN, Geneva.

• References

  1. Martin A., et al. preprint, http://xxx.arxiv.org/abs/math-ph/0510065 (2005).