Hyperboloid structures are architectural structures designed with hyperboloid geometry. Often these are tall structures such as towers where the hyperboloid geometry's structural strength is used to support an object high off the ground, but hyperboloid geometry is also often used for decorative effect as well as structural economy. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov (1853–1939).^{[1]} The world's first hyperboloid tower is located in Polibino, Lipetsk Oblast, Russia.
The shapes are doubly ruled surfaces (hence can be built with a lattice of straight beams), which can be classed as:
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Hyperbolic structures have a negative Gaussian curvature, meaning they curve inward rather than outward or being straight. As doubly ruled surfaces, they can be made with a lattice of straight beams, hence are easier to build and, all else equal, stronger than curved surfaces that do not have a ruling and must instead be built with curved beams.
Hyperboloid structures are superior in stability towards outside forces than "straight" buildings, but have shapes often creating large amounts of unusable volume (low space efficiency) and therefore are more commonly used as a purpose driven structure, such as water towers (to support a large mass), cooling towers, and aesthetic features, but their cross section is much more commonly seen in hyperbolic bridges.^{[2]}
With cooling towers, a hyperbolic structure is preferred. At the bottom, the widening of the tower provides higher surface area for water to boil in. As the water first boils and steam rises, the narrowing effect helps provide laminar, or straight vertical flow, and then as it widens out, turbulent flow is enabled and air is more easily mixed with outside air.^{[3]}
In the 1880s, Shukhov began to work on the problem of the design of roof systems to use a minimum of materials, time and labor. His calculations were most likely derived from mathematician Pafnuty Chebyshev's work on the theory of best approximations of functions. Shukhov's mathematical explorations of efficient roof structures led to his invention of a new system that was innovative both structurally and spatially. By applying his analytical skills to the doublycurved surfaces Nikolai Lobachevsky named "hyperbolic", Shukhov derived a family of equations that led to new structural and constructional systems, known as hyperboloids of revolution and hyperbolic paraboloids.
The steel gridshells of the exhibition pavilions of the 1896 AllRussian Industrial and Handicrafts Exposition in Nizhny Novgorod were the first publicly prominent examples of Shukhov's new system. Two pavilions of this type were built for the Nizhni Novgorod exposition, one oval in plan and one circular. The roofs of these pavilions were doublycurved gridshells formed entirely of a lattice of straight angleiron and flat iron bars. Shukhov himself called them azhurnaia bashnia ("lace tower", i.e., lattice tower). The patent of this system, for which Shukhov applied in 1895, was awarded in 1899.
Shukhov also turned his attention to the development of an efficient and easily constructed structural system (gridshell) for a tower carrying a large gravity load at the top – the problem of the water tower. His solution was inspired by observing the action of a woven basket holding up a heavy weight. Again, it took the form of a doublycurved surface constructed of a light network of straight iron bars and angleiron. Over the next twenty years, he designed and built close to two hundred of these towers, no two exactly alike, most with heights in the range of 12m to 68m.
At least as early as 1911, Shukhov began experimenting with the concept of forming a tower out of stacked sections of hyperboloids. Stacking the sections permitted the form of the tower to taper more at the top, with a less pronounced "waist" between the shapedefining rings at bottom and top. Increasing the number of sections would increase the tapering of the overall form, to the point that it began to resemble a cone.
By 1918 Shukhov had developed this concept into the design of a ninesection stacked hyperboloid radio transmission tower for Moscow. Shukhov designed a 350m tower, which would have surpassed the Eiffel Tower in height by 50m, while using less than a quarter of the amount of material. His design, as well as the full set of supporting calculations analyzing the hyperbolic geometry and sizing the network of members, was completed by February 1919; however, the 2200 tons of steel required to build the tower to 350m were not available. In July 1919, Lenin decreed that the tower should be built to a height of 150m, and the necessary steel was to be made available from the army's supplies. Construction of the smaller tower with six stacked hyperboloids began within a few months, and Shukhov Tower was completed by March 1922.
Antoni Gaudi and Shukhov carried out experiments with hyperboloid structures practically simultaneously, but independently, in 18801895. Antoni Gaudi used structures in the form of hyperbolic paraboloid (hypar) and hyperboloid of revolution in the Sagrada Família in 1910.^{[4]} In the Sagrada Família, there are a few places on the nativity facade  a design not equated with Gaudi's ruledsurface design, where the hyperboloid crops up. All around the scene with the pelican, there are numerous examples (including the basket held by one of the figures). There is a hyperboloid adding structural stability to the cypress tree (by connecting it to the bridge). The "bishop's mitre" spires are capped with hyperboloids.
In the Palau Güell, there is one set of interior columns along the main facade with hyperbolic capitals. The crown of the famous parabolic vault is a hyperboloid. The vault of one of the stables at the Church of Colònia Güell is a hyperboloid. There is a unique column in the Park Güell that is a hyperboloid. The famous Spanish engineer and architect Eduardo Torroja designed a thinshell water tower in Fedala^{[5]} and the roof of hippodrome "Zarzuela"^{[6]} in the form of hyperboloid of revolution. Le Corbusier and Félix Candela used hyperboloid structures (hypar).
A hyperboloid cooling tower was patented by Frederik van Iterson and Gerard Kuypers in 1918.^{[7]}
The Georgia Dome was the first HyparTensegrity dome to be built.^{[8]}
Cooling tower Puertollano, Spain 
A hyperboloid Kobe Port Tower, Kobe, Japan. 

Sagrada Família by Antoni Gaudi, hyperboloid detail of the facade, Barcelona, Spain 
Hyperboloid Cathedral in Brasilia by Oscar Niemeyer, Brazil 
Hyperboloid Ještěd Tower by Karel Hubáček, Czech Republic 
Partial view of the hyperbolic roof Olympiapark, Munich, Germany, tensile structure 
Hyperbolic paraboloid roof on Church Army Chapel, Blackheath, 1965 
