Non-Euclidean geometry and structural strength.
Consider the hollow sphere of metal. It’s strong, far stronger than a flat sheet of metal of the same thickness. There are several ways to explain this, one of these is that a sphere has a geometrical resistance to being bent. If a load is placed on the sphere, the part nearest the load is in compression and parts further away are in tension (known as hoop tension) and the sphere resists the load until the combination of compression and tension causes the material to compress and stretch enough for it finally to buckle. A sphere supports a very much higher buckling load than a plate or cylinder of the same thickness.
This geometric resistance to being bent is due to the sphere’s surface being a non-Euclidean geometry. A sphere’s surface is described by Riemennian geometry with positive curvature, any straight lines (ie. geodesics) forming a triangle on the surface will have angles at the corners that add up to more than 180 degrees.
A surface doesn’t have to be flat in order to have zero curvature. An open cylinder has zero curvature everywhere. In mathematical terms, every surface with flat curvature is known as ‘ruled’.
Given that flat surfaces are extremely weak, I find it startling and depressing how many structures are made from combinations of flat and cylindrical surfaces. Combining flat surfaces gives a structure that is stronger than individual flat surfaces, but still far weaker than surfaces with nonzero curvature.
It’s so rare for non-zero curvature to be used in structural design that it’s easily possible to list all the best examples. Domes in building construction were already in decline in the Gothic period from the 12th to 16th centuries. Modern geodesic domes are not the same, made to approximate a dome shape using flat panels. The TWA terminal in New York is one of the extremely few modern buildings that rely on non-Euclidean geometry for strength. Russian spacecraft such as Soyuz make great use of spherical shapes for strength. Car panels have a non-Euclidean shape, but even there in many cases the three-dimensional curvature of the shape is mere window dressing, rather than an essential part of the structural strength. The base of PET bottles uses both positive and negative curvature to achieve a high strengh not possible with a cylinder attached to a flat base.
Another less obvious example is diamond plate. This was originally designed as a cheap non-slip flooring, but it does rely on non-Euclidean geometry for strength. Diamond plate has the look to me of a product that was specifically done to avoid a patent. There are other more obvious combinations of bumps that give slightly stronger plates, usually based on tessellations, the important point for all strong arrangements of bumps is that there is no straight line on the surface along which the surface curvature in nearly zero. So, for instance, bumps must not be arranged in a triangular pattern.
So, how else can non-Euclidean geometry be used in construction? One simple example would be to create strong lightweight panels by spot-welding (or gluing) two diamond plates together with bumps touching bumps. Whether made of steel or aluminium, these made in different thicknesses would be suitable for a wide range of applications in making, for instance, lightweight floors of high-rise buildings and major parts of big trucks and other heavy machinery. For a non-structural application, note that these panels would make superb heat exchangers.
