A skyline, drawn by gravity.

Hold a loose chain by its two ends. Gravity arranges the links into a smooth curve called a catenary. It is tempting to call it a parabola, but those are different mathematical shapes. A chain finds its curve through the forces acting along it. Turn that image upside down and you have a useful starting point for thinking about an arch: a line discovered by weight can become a line that carries weight.

St. Louis's Gateway Arch has the form of an inverted catenary and rises 630 feet; its legs are also 630 feet apart at ground level. Eero Saarinen's apparently simple silhouette contains stacked, double-walled steel sections with triangular cross-sections. The polished exterior conceals an enormous amount of engineering. The real monument is not an ordinary chain flipped over and enlarged: its changing thickness, materials and loads belong to a carefully designed structure.

Visitors can travel to an observation deck at the top. That creates a less picturesque problem: how does an elevator follow a narrowing, curving leg? Designer Dick Bowser developed a special tram system rather than inserting a conventional vertical lift. The journey makes the geometry physical. Architecture is often photographed as a finished outline, but a good building also has to solve the routes, forces and maintenance hidden inside that outline. The next time a structure looks effortless, try sketching where its weight travels and where its people go.