Editing Gateway Arch (section)

===Mathematical elements===
[[File:St Louis Gateway Arch.jpg|thumb|left |The arch is a weighted catenary—its legs are wider than its upper section.]]
The geometric form of the structure was set by mathematical equations provided to Saarinen by the [[German-American]] engineer [[Hannskarl Bandel]]. Bruce Detmers and other architects expressed the geometric form in blueprints with this equation:<ref>{{cite web |url=https://www.nps.gov/jeff/planyourvisit/mathematical-equation.htm |title=Mathematical Equation |publisher=National Park Service |access-date=December 29, 2023 }}</ref>

<math>y = A \left( \cosh \frac {Cx}{L}-1 \right) \quad\Leftrightarrow\quad x = \frac {L}{C} \operatorname{arcosh} \left( 1 + \frac {y}{A} \right)</math>,

with the constants

<math>A = \frac {f_c} {Q_b/Q_t - 1} = 68.7672</math>

<math>C = \operatorname{arcosh} \frac {Q_b}{Q_t} = 3.0022</math>

where ''f<sub>c</sub>'' = {{cvt|625.0925|ft|m|4}} is the maximum height of centroid, ''Q<sub>b</sub>'' = {{cvt|1262.6651|sqft|m2|5}} is the maximum cross sectional area of arch at base, ''Q<sub>t</sub>'' = {{cvt|125.1406|sqft|m2|5}} is the minimum cross sectional area of arch at top, and ''L'' = {{cvt|299.2239|ft|m|5}} 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 function|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.<ref name="KabaiTóth">{{cite web |url=http://demonstrations.wolfram.com/JeffersonNationalExpansionMemorial/ |title=Jefferson National Expansion Memorial |last1=Kabai |first1=Sándor |last2=Tóth |first2=János |publisher=[[Wolfram Demonstrations Project]] |access-date=December 14, 2010 }}</ref><ref>{{cite web |last=Weisstein |first=Eric |author-link=Eric W. Weisstein |url=http://mathworld.wolfram.com/Catenary.html |title=Catenary |work=[[MathWorld]] }}</ref> 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.<ref name="Corrigan"/> The Gateway Arch itself is not a common catenary, but a more general curve of the form ''y''=''A''cosh (''Bx'').<ref>{{cite journal |last=Osserman |first=Robert |title=Mathematics of the Gateway Arch |url=https://www.ams.org/notices/201002/rtx100200220p.pdf |date=February 2010 |journal=[[Notices of the American Mathematical Society]] |issn=0002-9920 |volume=57 |issue=2 |pages=220–229 |archive-url=https://web.archive.org/web/20121023000925/http://www.ams.org/notices/201002/rtx100200220p.pdf |archive-date=October 23, 2012 |url-status=dead }}</ref> This makes it an ''inverted weighted catenary.''<ref name="The Rotarian">{{cite journal |url=https://books.google.com/books?id=DzcEAAAAMBAJ&q=weighted+catenary&pg=PA34 |title=Soaring Symbol for St. Louis |last=Hannon |first=Robert E. |pages=33–34 |journal=[[The Rotarian]] |date=June 1963 |volume=102 |issue=6 |issn=0035-838X }}</ref><ref>{{cite journal |url=https://books.google.com/books?id=BuMDAAAAMBAJ&q=weighted+catenary&pg=PA89 |title=The Incredible Gateway Arch: America's Mightiest National Monument |last=Hicks |first=Clifford B. |page=89 |journal=[[Popular Mechanics]] |date=December 1963 |volume=120 |issue=6 |issn=0032-4558 }}</ref> 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.<ref name="AIA 1983">{{cite journal |url=https://books.google.com/books?id=yQEyAQAAIAAJ |title=Is It a Catenary? |last=Crosbie |first=Michael J. |publisher=[[American Institute of Architects]] |date=June 1983 |volume=72 |issue=6 |journal=AIA Journal |pages=78–79 }}</ref>