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Pinched Torus
In mathematics, and especially topology and differential geometry, a pinched torus (or croissant surface) is a kind of two-dimensional surface. It gets its name from its resemblance to a torus that has been pinched at a single point. A pinched torus is an example of an orientable, compact 2-dimensional pseudomanifold. Parametrisation A pinched torus is easily parametrisable. Let us write . An example of such a parametrisation − which was used to plot the picture − is given by where: :f(x,y) = \left( g(x,y)\cos x , g(x,y)\sin x , \sin\!\left(\frac\right)\sin y \right) Topology Topologically, the pinched torus is homotopy equivalent to the wedge of a sphere and a circle. It is homeomorphic to a sphere with two distinct points being identified. Homology Let ''P'' denote the pinched torus. The homology groups of ''P'' over the integers can be calculated. They are given by: :H_0(P,\Z) \cong \Z, \ H_1(P,\Z) \cong \Z, \ \text \ H_2(P,\Z) \cong \Z. Cohomology The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinched Torus
In mathematics, and especially topology and differential geometry, a pinched torus (or croissant surface) is a kind of two-dimensional surface. It gets its name from its resemblance to a torus that has been pinched at a single point. A pinched torus is an example of an orientable, compact 2-dimensional pseudomanifold. Parametrisation A pinched torus is easily parametrisable. Let us write . An example of such a parametrisation − which was used to plot the picture − is given by where: :f(x,y) = \left( g(x,y)\cos x , g(x,y)\sin x , \sin\!\left(\frac\right)\sin y \right) Topology Topologically, the pinched torus is homotopy equivalent to the wedge of a sphere and a circle. It is homeomorphic to a sphere with two distinct points being identified. Homology Let ''P'' denote the pinched torus. The homology groups of ''P'' over the integers can be calculated. They are given by: :H_0(P,\Z) \cong \Z, \ H_1(P,\Z) \cong \Z, \ \text \ H_2(P,\Z) \cong \Z. Cohomology The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world. In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. 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 spindle 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. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientable Surface
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Surface
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudomanifold
In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain Mathematical singularity, singularities. For example, the cone of solutions of z^2=x^2+y^2 forms a pseudomanifold. A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities. The concepts of orientability, orientation and degree of a mapping make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts. Definition A topological space ''X'' endowed with a Triangulation (topology), triangulation ''K'' is an ''n''-dimensional pseudomanifold if the following conditions hold: # (''pure'') is the union (set theory), union of all ''n''-simplex, simplices. # Every is a Euclidean simplex#Faces, face of exactly one or two ''n''-simplices for ''n > 1''. # For every pair of ''n''-simplices σ and σ' in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedge Sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the quotient space of the disjoint union of ''X'' and ''Y'' by the identification x_0 \sim y_0: X \vee Y = (X \amalg Y)\;/, where \,\sim\, is the equivalence closure of the relation \left\. More generally, suppose \left(X_i\right)_ is a indexed family of pointed spaces with basepoints \left(p_i\right)_. The wedge sum of the family is given by: \bigvee_ X_i = \coprod_ X_i\;/, where \,\sim\, is the equivalence closure of the relation \left\. In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints \left(p_i\right)_, unless the spaces \left(X_i\right)_ are homogeneous. The wedge sum is again a pointed space, and the binary operation is associative and commuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
