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Domical Vault
In architecture, a cloister vault (also called a pavilion vault) is a vault with four concave surfaces (patches of cylinders) meeting at a point above the center of the vault. It can be thought of as formed by two barrel vaults that cross at right angles to each other: the open space within the vault is the intersection 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, 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 of a cloister vault is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Section (geometry)
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation. In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally crosshatched with the style of crosshatching often indicating the types of materials being used. With computed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 is called a bicylinder. Topologically, it is equivalent to a square hosohedron. The intersection of three cylinders is called a tricylinder. A bisected bicylinder is called a vault, and a cloister vault in architecture has this shape. Steinmetz solids are named after mathematician Charles Proteus Steinmetz, who solved the problem of determining the volume of the intersection. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. Bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac r^3 and the ;surface area :A=16 r^2. The upper half of a bicylinder is the square ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 is chiefly appreciated for his art. His painting is characterized by its serene humanism, its use of geometric forms and perspective. His most famous work is the cycle of frescoes ''The History of the True Cross'' in the church of San Francesco in the Tuscan town of Arezzo. Biography Early years Piero was born Piero di Benedetto in the town of Borgo Santo Sepolcro, modern-day Tuscany, to Benedetto de' Franceschi, a tradesman, and Romana di Perino da Monterchi, members of the Florentine and Tuscan Franceschi noble family. His father died before his birth, and he was called Piero della Francesca after his mother, who was referred to as "la Francesca" due to her marriage into the Franceschi family (similar to how Lisa Gherardini became kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 3.1415927, a record in accuracy which would not be surpassed for over 800 years. Life and works Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravages of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang () at one point held the position of Chief Minister for the Palace Buildings () within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuozhi (), also served the court and was greatly respected for his erudition. Zu was born in Jiankang. His family had historically been involved in astronomical research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his tal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including (ε, δ)-definition of limit, codify ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. * 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 definition of length (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. In ancient times, volume is measured using similar-shaped natural containers and later on, standardized containers. Some simple three-dimensional shapes can have its volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal vo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: Laurence King Publishing, p. 949. The word "groin" refers to the edge between the intersecting vaults. Sometimes the arches of groin vaults are pointed instead of round. In comparison with a barrel vault, a groin vault provides good economies of material and labor. The thrust is concentrated along the groins or arrises (the four diagonal edges formed along the points where the barrel vaults intersect), so the vault need only be abutted at its four corners. Groin vault construction was first exploited by the Romans, but then fell into relative obscurity in Europe until the resurgence of quality stone building brought about by Carolingian and Romanesque architecture. It was superseded by the more flexible rib vaults of Gothic architectur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rome Synagogue Dome View From The Hill Aventine
, established_title = Founded , established_date = 753 BC , founder = King Romulus (legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption = The territory of the ''comune'' (''Roma Capitale'', in red) inside the Metropolitan City of Rome (''Città Metropolitana di Roma'', in yellow). The white spot in the centre is Vatican City. , pushpin_map = Italy#Europe , pushpin_map_caption = Location within Italy##Location within Europe , pushpin_relief = yes , coordinates = , coor_pinpoint = , subdivision_type = Country , subdivision_name = Italy , subdivision_type2 = Region , subdivision_name2 = Lazio , subdivision_type3 = Metropolitan city , subdivision_name3 = Rome Capital , government_footnotes= , government_type = Strong Mayor–Council , leader_title2 = Legislature , leader_name2 = Capitoline Assembl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero (0) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Union of two sets The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, :A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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