Catenary arch
A catenary arch is a type of architectural pointed arch that follows an inverted catenary curve. It is very common in nature, cathedrals and in Gothic arches used in Gothic architecture.It is not a parabolic arch.
A natural bridge retaining a catenary profile after several million years of erosion by water and wind, optimally fitting itself to the strongest curve possible.
A sustainable construction method making use of the centenary arch/curve known as Timbrel vaulting or Catalan vaulting or Gusatavino vaulting in the USA, was refined in the 14th century around the Mediterranean, although its precise origins are unknown people have been building in this technique for thousands of years in upper Egypt.
Early geomimicry in Africa
Further refinements in Africa
Image source
The resulting vault does not rely on gravity but on the adhesion of several layers of overlapping terracotta tiles which are woven together with fast-setting mortar, hence is self-supporting apart from some temporarily required light form-work. Adding two or three layers makes the resulting laminated shell almost as strong as reinforced concrete.
Golden times in Europe; 1600s'
Guastavino; the satoshi of thin vaults, pictured standing on a vault below Introduced Catalan vault to America
Convincing the public in USA; 1900s'
The result defies common sense, because a timbrel vault is very thin compared to a Roman vault, while at the same time it is capable of bearing much higher loads. This of course enables wider spans and gentler curves.
New Era begins, Guastavino builds thin vaults in USA; 1900s'
New Era begins again, The millenium and beyond; computers push the limits of the thin arch; 2000s'
References
https://en.wikipedia.org/wiki/Catalan_vault
http://www.block.arch.ethz.ch/brg/project/free-form-catalan-thin-tile-vault