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The catenoid is a classic minimal surface formed by rotating a catenary curve—defined by the equation y = cosh(x)—about its horizontal axis. What makes it mathematically elegant is that it has zero mean curvature everywhere, making it one of the simplest examples of a nontrivial minimal surface. Structurally, it resembles a smooth, hourglass-like shape, and it can be physically observed in soap films stretched between two circular rings. Unlike most other surfaces, the catenoid balances surface
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