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In 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 construction, constructi ...
, 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 (enclosur ...
with four convex surfaces (patches of cylinders
A cylinder () 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 infinite ...
) 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 ...
of the space within the two barrel vaults, and the solid material that surrounds the vault is the union 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: La ...
, 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 base. Domical vaults can have other polygons as cross-sections (especially octagons) rather than being limited to squares.
Geometry
Any horizontalcross-section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture and engineering 3D
* Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
...
of a cloister vault is a square. This fact may be used to find the volume
Volume is a measure of regions in 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) ...
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 is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
students, and was solved long ago by Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
in Greece, Zu Chongzhi
Zu Chongzhi (; 429 – 500), courtesy name Wenyuan (), was a Chinese astronomer, inventor, mathematician, politician, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1415 ...
in China, and Piero della Francesca
Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
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 where s is the length of the side of the square base.
See also
*List of architectural vaults
The following is a list of arched structures known in architecture as Vault (architecture), vaults.
* Annular vault – A Barrel vault springing from two concentric walls.
* Barrel vault – An architecture tunnel vault or barrel vault is a semic ...
References
{{reflist Arches and vaults