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Real Projective Plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in passing through the origin. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a square by glui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boy's Surface
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space. Boy's surface was first parametrized explicitly by Bernard Morin in 1978. Another parametrization was discovered by Rob Kusner and Robert Bryant.. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. Unlike the Roman surface and the cross-cap, it has no other singularities than self-intersections (that is, it has no pinch-points). Symmetry of the Boy's surface Boy's surface has 3-fold symmetry. This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking exactly the same. The Boy's surface can be cut into three mutually congruent pieces. Model at Oberwolfach The Mathematical Research Institute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, is an immersion if :D_pf : T_p M \to T_N\, is an injective function at every point ''p'' of ''M'' (where ''TpX'' denotes the tangent space of a manifold ''X'' at a point ''p'' in ''X''). Equivalently, ''f'' is an immersion if its derivative has constant rank equal to the dimension of ''M'': :\operatorname\,D_p f = \dim M. The function ''f'' itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion that is also a topological embedding, so that ''M'' is diffeomorphic to its image in ''N''. An immersion is precisely a local embedding – that is, for any point there is a neighbourhood, , of ''x'' such that is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Symmetry Group Ci
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the 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 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), and some point in it, and all the loops both starting and ending at this point— 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 along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Map
A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete space D and for every x \in X an open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a homeomorphism for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is connected. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the cardinality of the space D. If E is path-connected, then the covering \pi : E \rightarrow X is denoted as a path-connected covering. Examples * For every topological space X there exists the covering \pi:X \rightarrow X with \pi(x)=x, which i ... [...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] |
Conditional Clause
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is ''conditional'' on the dependent clause. A full conditional thus contains two clauses: a dependent clause called the ''antecedent'' (or ''protasis'' or ''if-clause''), which expresses the condition, and a main clause called the ''consequent'' (or ''apodosis'' or ''then-clause'') expressing the result. Languages use a variety of grammatical forms and constructions in conditional sentences. The forms of verbs used in the antecedent and consequent are often subject to particular rules as regards their tense, aspect, and mood. Many languages have a specialized type of verb form called the conditional mood – broadly equivalent in meaning to the English "would (do something)" – for use in some types of conditional sentences. Types of conditiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antipodal Point
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true diameter). This term applies to opposite points on a circle or any n-sphere. An antipodal point is sometimes called an antipode, a back-formation from the Greek loan word ''antipodes'', meaning "opposite (the) feet", as the true word singular is ''antipus''. Theory In mathematics, the concept of ''antipodal points'' is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite ''through the centre''; for example, taking the centre as origin, they are points with related vectors v and −v. On a circle, such points are also called diametrically opposite. In other words, each line through the centre intersects the sphere in two points, one for each ray out from the centre, and these two poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the ''minor arc'', and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
