It’s not quite precisely phrased, but this is just a simple literal mathematical minimum - like that a sphere is the surface which uses the least material to en

It’s not quite precisely phrased, but this is just a simple literal mathematical minimum - like that a sphere is the surface which uses the least material to enclose a given volume. A cooling tower needs to be a chimney - a closed off area at ground level, open at a higher altitude. Given those constraints alone, the cooling tower form is literally the one that minimizes the amount of material required. A circle uses less material to enclose the same area of the ground than any other shape; the cooling tower’s curved form uses less material than any other shape to create any area of opening at any height above the ground. It might seem intuitive that a cylinder would be more efficient - if you imagine building one vertical slice of the tower wall, going in a straight line up seems like it would require less bricks than curving inwards then back out. But if you build one layer at a time, it’s clear that if you can build smaller and smaller circles before getting wider again nearer the top you will use less bricks than if you just keep building every layer with a circle of the full diameter. If you balance both of these ‘stretching’ directions - minimize the size of each circle but also minimize the path from each point at the base to a point at the top - you get a cooling-tower type surface. Actually this mathematical ‘minimal surface’ between two rings - as in the way to create two walls that are circular at the base and the top using the minimum amount of material - is not a hyperboloid, but a catenoid (the curve is a catenary not a paraboloid). The surface does exhibit mixed curvature though, so it is ‘hyperbolic’ if not ‘hyperboloid’. The easiest way to see the catenoid as a minimal surface is with a soap film - it is the shape that forms when a soap bubble is stretched between two rings. Surface tension forces soap films to minimize area. In practice cooling towers have other constraints than minimizing material cost - structural, wind load, material strength, ease of construction, etc, so they might actually be a different shape than a pure catenoid. But it is true that the closer to a catenoid they are, the less material they will use - and that is one good reason why they are not cylinders or truncated cones or pyramids or barrel shapes, and as a result they do have negative Gaussian curvature.