page_title: Gateway Arch
section_title: Mathematical elements
hierarchical_section_title: Gateway Arch / Characteristics / Mathematical elements
caption_attribution_description: Gateway Arch, St. Louis, Missouri English: The Gateway Arch, part of the Jefferson National Expansion Memorial in St. Louis, Missouri, framing the courthouse where the Dred Scott decision was read. Deutsch: Gateway Arch in St. Louis
context_page_description: The Gateway Arch is a 630-foot monument in St. Louis, Missouri, United States. Clad in stainless steel and built in the form of a weighted catenary arch, it is the world's tallest arch, the tallest man-made monument in the Western Hemisphere, and Missouri's tallest accessible building. Built as a monument to the westward expansion of the United States, and officially dedicated to "the American people," the Arch, commonly referred to as "The Gateway to the West" is the centerpiece of Gateway Arch National Park and has become an internationally recognized symbol of St. Louis, as well as a popular tourist destination.
  The Arch was designed by Finnish-American architect Eero Saarinen in 1947; construction began on February 12, 1963 and was completed on October 28, 1965 at an overall cost of $13 million. The monument opened to the public on June 10, 1967.
  It is located at the site of St. Louis's founding on the west bank of the Mississippi River.
context_section_description: The geometric form of the structure was set by mathematical equations provided to Saarinen by Hannskarl Bandel. Bruce Detmers and other architects expressed the geometric form in blueprints with this equation:
  ,
  with the constants
  where fc = 625.0925 ft (191 m) is the maximum height of centroid, Qb = 1,262.6651 sq ft (117 m²) is the maximum cross sectional area of arch at base, Qₜ = 125.1406 sq ft (12 m²) is the minimum cross sectional area of arch at top, and L = 299.2239 ft (91 m) is the half width of centroid at the base. The triangular cross sectional area varies linearly with the vertical height of its centroid.
  This hyperbolic cosine function describes the shape of a catenary. A chain that supports only its own weight forms a catenary; the chain is purely in tension. Likewise, an inverted catenary arch that supports only its own weight is purely in compression, with no shear. The catenary arch is the stablest of all arches since the thrust passes through the legs and is absorbed in the foundations, instead of forcing the legs apart. The Gateway Arch itself is not a common catenary, but a more general curve of the form y=Acosh (Bx). This makes it an inverted weighted catenary. Saarinen chose a weighted catenary over a normal catenary curve because it looked less pointed and less steep. In 1959, he caused some confusion about the actual shape of the arch when he wrote, "This arch is not a true parabola, as is often stated. Instead it is a catenary curve—the curve of a hanging chain—a curve in which the forces of thrust are continuously kept within the center of the legs of the arch." William V. Thayer, a professor of mathematics at St. Louis Community College, later wrote to the St. Louis Post-Dispatch calling attention to the fact that the structure was a weighted catenary.
caption: The arch is a weighted catenary—its legs are wider than its upper section.
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