|
Solid Torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a ''solid torus'' include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels. Topological properties The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to S^1 \times S^1, the ordinary torus. Since the disk D^2 is contractible, the solid torus has the homotopy type o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a Lemon (geometry), spindle torus (or ''self-crossing torus'' or ''self-intersecting torus''). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bagel
A bagel (; ; also spelled beigel) is a bread roll originating in the Jewish communities of Poland. Bagels are traditionally made from yeasted wheat dough that is shaped by hand into a torus or ring, briefly boiled in water, and then baked. The result is a dense, chewy, doughy interior with a browned and sometimes crisp exterior. Bagels are often topped with seeds baked on the outer crust—traditional choices include poppy and sesame seeds—or with salt grains. Different dough types include whole-grain and rye. The basic roll-with-a-hole design, hundreds of years old, allows even cooking and baking of the dough; it also allows groups of bagels to be gathered on a string or dowel for handling, transportation, and retail display. The earliest known mention of a boiled-then-baked ring-shaped bread can be found in a 13th-century Syrian cookbook, where they are referred to as . Bagel-like bread known as obwarzanek was common earlier in Poland as seen in royal family account ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Reeb Foliation
In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf. By Novikov's compact leaf theorem, every smooth foliation of the 3-sphere includes a compact torus leaf, bounding a solid torus foliated in the same way. Illustrations References * * * External links * {{topology-stub Foliations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hyperbolic Dehn Surgery
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions. Such an operation is often also called hyperbolic Dehn filling, as Dehn surgery proper refers to a "drill and fill" operation on a link which consists of ''drilling'' out a neighborhood of the link and then ''filling'' back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling". We will generally assume that a hyperbolic 3-manifold is complete. Suppose ''M'' is a cusped hyperbolic 3-manifold with ''n'' cusps. ''M'' can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cheerios
Cheerios is a brand of cereal manufactured by General Mills in the United States and Canada, consisting of pulverized oats in the shape of a solid torus. In Europe, Cheerios is marketed by Cereal Partners under the Nestlé brand; in Australia and New Zealand, Cheerios is sold as an Uncle Tobys product. It was first manufactured in 1941 as CheeriOats. History Cheerios were introduced on May 1, 1941, as "Cheerioats". The name was shortened to "Cheerios" in 1945, after a competing cereal manufacturer, Quaker Oats, claimed to hold the rights to use the term "oats". Cheerios' production was based upon the extrusion process invented for Kix in 1937. The oat flour process starts in Minneapolis before being shipped to factories in Iowa, Georgia, and Buffalo, New York. On 1976, 35 years after the cereal was first introduced, "Cinnamon Nut Cheerios" became the first alternate variety of Cheerios to be sold in stores. Nearly 3 years later, in 1979, " Honey Nut Cheerios" was intr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Homology (mathematics)
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of Abelian group, abelian groups called ''homology groups.'' This operation, in turn, allows one to associate various named ''homologies'' or ''homology theories'' to various other types of mathematical objects. Lastly, since there are many homology theories for Topological space, topological spaces that produce the same answer, one also often speaks of the ''homology of a topological space''. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a Cochain complexes, cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fundamental Group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have Group isomorphism, isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface (mathematics), surface), and some point in it, and all the loops both starting and ending at this point—path (topology), paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then alo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, '' The Daily Princetonian'', and later added book publishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a nonempty topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any path-connected space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
