Gauss’s “Remarkable Theorem” and the best way to hold a pizza slice (2014)

I guess that to build it "for its required purpose" is implicit in the sentence. I agree, that the author could have gone into more detail of what are the requirements of cooling towers but that may be out of the scope of the article.

It is claimed here [1] that the hyperboloid shape is not just for structural efficiency, but also for aerodynamic reasons, efficiently generating an updraft from a relatavely small temperature difference.

They are not just for nuclear plants, but for any large steam generating facility lacking a large source of cooling water in its local environment.

[1] https://en.wikipedia.org/wiki/Hyperboloid_structure

https://en.wikipedia.org/wiki/Hyperboloid#/media/File:Cylind...

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.

I fail to see how what you’re saying negates the article. In fact, there are several descriptions of folding as being the next step (see folded paper holding a can of beans).

assuming the insistence on folding to be a sign of nyc-ness, is your take on pizza so nyc-centric that it is a local cuisine of the tri-borough area?