Stable tilt.
Twenty-three centuries ago, Archimedes laid the foundations for modern shipbuilding with his treatise On Floating Bodies. Among other things, he proved that a buoy shaped like a spherical bowl can float stably in only two positions: right-side up or upside down. A bullet-shaped buoy with a parabolic bottom and a flat top, however, can float stably in a tipped position.
But Archimedes, having neither calculus nor computers, wasn't able to answer the next question: Can a parabolic buoy float with such an extreme tilt that its top is partially submerged? Now, Chris Rorres, a retired math professor at Drexel University, has written what might be called the last chapter of Archimedes' book.
In this summer's issue of Mathematical Intelligencer, he shows that a paraboloid with a partly submerged top can be stable in as many as three positions--if the object can't decide whether to behave like a wide bowl or a narrow rod. Using a 35-year-old branch of mathematics called catastrophe theory, Rorres pinned down exactly where this identity crisis begins: For a buoy with a density of 0.5, for example, it's when the height reaches  35/48, or about 0.85 times the width. Above this magic number, if the density or shape vary a little, the paraboloid can suddenly--or "catastrophically"--tumble over.
Horst Nowacki, a naval architect at the Technical University of Berlin, observes that catastrophe theory is a "new tool for the naval community." Rorres says the math may help predict the behavior of icebergs, which sometimes turn turtle as they melt.
CREDIT: C. RORRES |
