A lot of major advances of science that are used by laypeople every day are applications of such tangible fields as physics, chemistry, or biology, but there’s one major component of the modern civilization that is based on pure math. It is a bridge, a suspension bridge to make it the best example, something millions of people trust their lives to, every single day, even if they believe mathematics to be useless.
Of course, the humans have been building bridges since prehistoric times, enabling transportation, which, in turn, enabled trade and economy and the progress. Those who build a bridge have to account for a great variety of forces that will act on it during its lifetime: the weight of the bridge itself, the weight of the traffic it is expected to support, the environmental factors of wind, earthquake, or water. All of that combines into tension, compression, shear, and torsion, and different materials handle these differently. Stone handles compression but not tension or torsion. Wood handles tension better. Ropes and cables handle tension and torsion, but not compression. Steel handles everything, but is heavy and expensive.
One of the bridge type, as old as mankind itself, is the suspension bridge. First made of vines and ropes that are thrown across a chasm, with wooden plank decks later added, such bridges worked in situations where no solid stone arch or wooden beam or cantilever or any other engineering structure would be feasible. One such suspension bridge built in the 7th century in Mayan city of Yaxchilan remained the longest bridge in the world until the masonry arch Trezzo Bridge was completed in Lomardy in 1377. Modern suspension bridges, however, are not catenary, which means the shape of the bridge deck is not following the catenary curve, but is instead straight or even slightly arched upwards, strong enough to let vehicles and even trains to cross. The deck is suspended on a multitude of cables, which are either directly attached to the pillars (a construction known as cable-stayed bridge) or are vertical, attached to the main cables which are catenary (the commonly seen suspension bridge design). The weight of the deck turns onto tension load on the cables and compression load on the pillars where the main cables are connected.
The precise geometries of the cables strung in the air have given the suspension bridges a look of mathematical perfection and turned some of them from utility construction into major landmarks and tourist attractions. Their pictures appear in glossy magazines and coffee table books instead of engineering textbooks. It is also impressive to see how the line of technology is unbroken for thousands of years, from the first vine across a chasm to the colossal modern day bridges; what changed are the materials, the precision, and the daring reach of science and engineering defying nature.