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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] |
Bernard Morin
Bernard Morin (; 3 March 1931 in Shanghai, China – 12 March 2018) was a French mathematician, specifically a topologist. Early life and education Morin lost his sight at the age of six due to glaucoma, but his blindness did not prevent him from having a successful career in mathematics. He received his Ph.D. in 1972 from the Centre National de la Recherche Scientifique. Career Morin was a member of the group that first exhibited an eversion of the sphere, i.e., a homotopy which starts with a sphere and ends with the same sphere but turned inside-out. He also discovered the Morin surface, which is a half-way model for the sphere eversion, and used it to prove a lower bound on the number of steps needed to turn a sphere inside out. Morin discovered the first parametrization of Boy's surface (earlier used as a half-way model), in 1978. His graduate student François Apéry, in 1986, discovered another parametrization of Boy's surface, which conforms to the general method for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thurston Sphere Eversion
Thurston may refer to: Places Antarctica *Thurston Glacier, Marie Byrd Land, Antarctica *Thurston Island, off Ellsworth Land, Antarctica United Kingdom * Thurston, Suffolk, England, a village **Thurston railway station *Thurston's Hall, a former snooker and billiards venue in London * Thurston End, a hamlet in Hawkedon parish United States *Thurston County, Nebraska **Thurston, Nebraska, a village *Thurston, New York, a town *Thurston, Ohio, a village *Thurston, Oregon (other), several places * Thurston, Virginia, an unincorporated community *Thurston County, Washington People *Thurston (name), a list of people with this given name or surname Other uses *Thurston Gardens, botanical gardens in Suva, Fiji *Thurston Elementary School, Ann Arbor, Michigan *Thurston High School, Springfield, Oregon *Thurston House (other), several houses on the US National Register of Historic Places * USS ''Thurston'' (AP-77), a World War II troop transport See also *Thurstone *Th ... [...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] |
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] |
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] |
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] |
Half-way Model
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] |
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] |
Mathematical Visualization
Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it most frequently consists of using computers to make static two or three dimensional drawings, animations, or interactive programs. Writing programs to visualize mathematics is an aspect of computational geometry. Applications Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves, space curves, polyhedra, ordinary differential equations, partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films), conformal maps, fractals, and chaos. Geometry Geometry can be defined as the study of shapes their size, angles, dimensions and proportions Linear algebra Complex analysis In c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiefel Manifold
In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold V_k(\Complex^n) of orthonormal ''k''-frames in \Complex^n and the quaternionic Stiefel manifold V_k(\mathbb^n) of orthonormal ''k''-frames in \mathbb^n. More generally, the construction applies to any real, complex, or quaternionic inner product space. In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent ''k''-frames in \R^n, \Complex^n, or \mathbb^n; this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group. Top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-principle
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas. The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric ''C''1 embedding theorem and the Smale–Hirsch immersion theorem. Rough idea Assume we want to find a function ''ƒ'' on R''m'' which satisfies a partial differential equation of degree ''k'', in co-ordinates (u_1,u_2,\dots,u_m). One can rewrite it as :\Psi(u_1,u_2,\dots,u_m, J^k_f)=0 where J^k_f stands for all parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
