Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

June 9, 2007

Advanced Floating

You've probably heard the one about Archimedes, the Greek mathematician who jumped out of the tub and ran naked through the streets shouting, "Eureka! Eureka!" Legend has it he suddenly realized how to tell a wreath made of pure gold from one made of a mixture of gold and silver. First, put a weight of pure gold equal to the wreath's weight into a bowl and fill the bowl with water. Then, take out the weight and put the wreath into the bowl; if the wreath contained the lighter and bulkier element silver, the bowl would overflow.

That story probably isn't true. Archimedes made many far more important discoveries during his lifetime—discoveries that would have excited him a good deal more than water spilled out of a tub. But it does have an element of truth: Archimedes was interested in things that float, so much so that he wrote two books about them.

He was especially fascinated by paraboloids (see below), objects that look like rocket nosecones. How do you think a paraboloid would float? With its nose down and flat side up? On its side? At some angle? With its flat side down?

Archimedes found that a paraboloid could float in different ways. Which way if floated depended on two things: its density (its weight compared to the weight of an equivalent volume of the liquid it was floating in) and the paraboloid's taper.

Archimedes looked only at cases where the paraboloid's flat base is completely above or below the liquid's surface, because that is all the math of his day could handle. Recently, mathematician Chris Rorres of the University of Pennsylvania completed the analysis by working out what happens when the base is partially submerged.

A paraboloid almost as dense as the liquid would float point down, with its flat base sticking up out of the liquid a little. But if its surface eroded and it became narrower, it might suddenly tilt and tumble onto its side.

The same thing can happen to icebergs, many of which float with broad bases up and points under water. As the iceberg melts, its underwater part is whittled down, until suddenly the entire mass topples over, throwing off any penguins or polar bears who happen to be hitching a ride.

A paraboloid much less dense than the liquid will float upright, with its base barely immersed. But if the liquid somehow became less dense, this paraboloid might fall over, too.

This also happens in real life. During an earthquake, a tall building built on water-saturated soil is like a floating paraboloid. As the soil "liquefies" and becomes less dense, the building can tilt, exposing its base, and then fall over.
Archimedes would have loved it.

You can see animations of floating paraboloids at

Muse, January 2005, p. 23.

Images courtesy of Chris Rorres.

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