Rome synagogue dome view from the hill Aventine

architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...

, a cloister vault (also called a pavilion vault) is a

vault Vault may refer to: * Jumping, the act of propelling oneself upwards Architecture * Vault (architecture), an arched form above an enclosed space * Bank vault, a reinforced room or compartment where valuables are stored * Burial vault (enclosure ...

with four concave surfaces (patches of

cylinders A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infini ...

) meeting at a point above the center of the vault. It can be thought of as formed by two

barrel vault A barrel vault, also known as a tunnel vault, wagon vault or wagonhead vault, is an architectural element formed by the extrusion of a single curve (or pair of curves, in the case of a pointed barrel vault) along a given distance. The curves are ...

s that cross at right angles to each other: the open space within the vault is the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of the space within the two barrel vaults, and the solid material that surrounds the vault is the

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

of the solid material surrounding the two barrel vaults. In this way it differs from a

groin vault A groin vault or groined vault (also sometimes known as a double barrel vault or cross vault) is produced by the intersection at right angles of two barrel vaults. Honour, H. and J. Fleming, (2009) ''A World History of Art''. 7th edn. London: L ...

, which is also formed from two barrel vaults but in the opposite way: in a groin vault, the space is the union of the spaces of two barrel vaults, and the solid material is the intersection. A cloister vault is a square domical vault, a kind of vault with a polygonal cross-section. Domical vaults can have other polygons as cross-sections (especially octagons) rather than being limited to squares.

Geometry

Any horizontal

cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Ab ...

of a cloister vault is a square. This fact may be used to find the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of the vault using

Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

. Finding the volume in this way is often an exercise for first-year

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

students, and was solved long ago by

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

in Greece,

Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3. ...

in China, and

Piero della Francesca Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...

in Renaissance Italy;. for more, see

Steinmetz solid In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders ...

. Assuming the intersecting barrel-vaults are semi-cylindrical, the volume of the vault is

\fracs^3

where s is the length of the side of the square base.

References

{{reflist Arches and vaults

Geometry

See also

References