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Minimax Eversion
In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models. It is a variational method, and consists of special homotopies (they are shortest paths with respect to Willmore energy); contrast with Thurston's corrugations, which are generic. The original method of half-way models was not optimal: the regular homotopies passed through the midway models, but the path from the round sphere to the midway model was constructed by hand, and was not gradient ascent/descent. Eversions via half-way models are called ''tobacco-pouch eversions'' by Francis and Morin. Half-way models A half-way model is an immersion of the sphere S^2 in \R^3, which is so-called because it is the half-way point of a sphere eversion. This class of eversions has time symmetry: the first half of the regular homotopy goes from the standard round sphere to the half-way model, and the second half (which goes from the half-way model to the inside-out sphere) is the same proces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word '' eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let :f\colon S^2\to \R^3 be the standard embedding; then there is a regular homotopy of immersions :f_t\colon S^2\to \R^3 such that ''ƒ''0 = ''ƒ'' and ''ƒ''1 = −''ƒ''. History An existence proof for crease-free sphere eversion was first created by . It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willmore Energy
In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore. Definition Expressed symbolically, the Willmore energy of ''S'' is: : \mathcal = \int_S H^2 \, dA - \int_S K \, dA where H is the mean curvature, K is the Gaussian curvature, and ''dA'' is the area form of ''S''. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic \chi(S) of the surface, so : \int_S K \, dA = 2 \pi \chi(S), which is a topological invariant and thus independent of the particular embedding in \mathbb^3 that was chosen. Thus the Willmore energy can be expressed as : \mathcal = \int_S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax Sphere Eversion
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. Game theory In general games The maximin value is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is: :\underline = \max_ \min_ Where: * is the index of the player of interest. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rob Kusner
Rob or ROB may refer to: Places * Rob, Velike Lašče, a settlement in Slovenia * Roberts International Airport (IATA code ROB), in Monrovia, Liberia People * Rob (given name), a given name or nickname, e.g., for Robert(o), Robin/Robyn * Rob (surname) * ''Rob.'', taxonomic author abbreviation for William Robinson (gardener) (1838–1935), Irish practical gardener and journalist Fictional characters * Rob, a character from the Cartoon Network series '' The Amazing World of Gumball'' * ROB 64, a character in the ''Star Fox'' video game series Arts, entertainment, and media Gaming * '' Castlevania: Rondo of Blood'', a 1993 video game nicknamed ''Castlevania: ROB'' * R.O.B., an accessory for the Nintendo Entertainment System Reports * ''ISM Report On Business'' (informally, "The R.O.B."), an economic report issued by the Institute for Supply Management * ''Report on Business'', or "ROB", a section of the ''Globe and Mail'' newspaper Other uses in arts, entertainment, and ... [...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] |
Saddle Point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willmore Flow
In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore. Definition Expressed symbolically, the Willmore energy of ''S'' is: : \mathcal = \int_S H^2 \, dA - \int_S K \, dA where H is the mean curvature, K is the Gaussian curvature, and ''dA'' is the area form of ''S''. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic \chi(S) of the surface, so : \int_S K \, dA = 2 \pi \chi(S), which is a topological invariant and thus independent of the particular embedding in \mathbb^3 that was chosen. Thus the Willmore energy can be expressed as : \mathcal ... [...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] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morin Surface
The Morin surface is the half-way model of the sphere eversion discovered by Bernard Morin. It features fourfold rotational symmetry. If the original sphere to be everted has its outer surface colored green and its inner surface colored red, then when the sphere is transformed through homotopy into a Morin surface, half of the outwardly visible Morin surface will be green, and half red: ''Half of a Morin surface corresponds to the exterior (green) of the sphere to which it is homeomorphic, and the other symmetric half to the interior (red).'' Then, rotating the surface 90° around its axis of symmetry will exchange its colors, i.e. will exchange the inner-outer polarity of the orientable surface, so that retracing the steps of the homotopy at exactly the same position back to the original sphere after having so rotated the Morin surface will yield a sphere whose outer surface is red and whose inner surface is green: a sphere which has been turned inside out. The following is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
