Catenary
A catenary is the shape assumed by a hanging chain. More precisely, it is the solution of the following mathematical problem: of all plane curves of a fixed length joining two fixed points, which has the least potential energy in a uniform gravitational field? (Or, alternatively, which curve has the lowest center of mass?) If the two fixed endpoints are at the same height, the catenary joining them looks roughly like a U or a parabola; however, it is not a parabola. While parabolas have the shape of the curve y = x2, catenaries have the shape of the graph of the hyperbolic cosine function, y = cosh(x).
Curiously, catenaries also arise in an unrelated problem: if two circles lie in parallel planes, both centered on an axis perpendicular to the planes, what is the surface of least area joining them? The surface can be found experimentally by making the circles out of wire and dipping them in a soap solution. When the circles are removed, the soap will form a film with the two circles as boundary, and this film will have the least possible area because that is the configuration with the least surface energy. If the two circles are far apart, the soap will simply form two separate disks. But if the circles are close together, the soap will form a concave tube linking the two circles. This tube, called a catenoid, has the same shape as a catenary rotated about the central axis. [See figure.]
When a surface has less area than any other surface with the same boundary, it is called a minimal surface. Catenoids were among the first minimal surfaces to be known by an explicit formula, and they are the only minimal surfaces of revolution. The study of minimal surfaces has been an active area of mathematical research for more than 150 years, and many new minimal surfaces have been discovered in recent years thanks to computer graphics.
Catenaries also are useful in solving a more whimsical problem of recent vintage. What kind of road will allow a square wheel to roll smoothly? To accomplish this, the axle must move along a horizontal line even though the road itself is not level. The solution is a road made of cobblestones in the shape of upside-down catenaries, joined together at 90-degree angles. The side length of the square wheel is the same as the arc length of each catenary "cobblestone."
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