In geometry , the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature . Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid. TOC Theoretical Pseudosphere In its general interpretation, a pseudosphere of radius R is any surface of Gaussian curvature curvature &minus 1 R sup 2 sup precisely, a complete metric space complete , simply connected surface of that curvature , by analogy with the sphere of radius R , which is a surface of curvature 1 R sup 2 sup . The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry . ref Citation first Eugenio last Beltrami title Saggio sulla interpretazione della geometria non euclidea journal Gior. Mat. volume 6 pages 248&ndash 312 language Italian year 1868 br Also Citation first Eugenio last Beltrami title Opere Matematiche volume 1 pages 374&ndash 405 language Italian isbn 1418184349 br ... revolving a tractrix about its asymptote . As an example, the half pseudosphere with radius 1 is the surface ... pseudosphere comes about because it is a dimension two dimensional surface of constant negative ... a positive number positively curved geometry of a dome the whole pseudosphere has at every point the negative ... Huygens found that the volume and the surface area of the pseudosphere are finite, ref cite book title ... ref ref MathWorld title Pseudosphere urlname Pseudosphere ref Universal covering space The half pseudosphere ... of the pseudosphere and the vertical geodesics x c to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion y 1 of the upper half plane as the universal covering space of the pseudosphere. The precise mapping ... model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere . ref Citation first ... Interactive demonstration of the pseudosphere at the University of Manchester Category Differential ... more details
could be realized on a surface of constant negative Gaussian curvature curvature , a pseudosphere . For Beltrami s concept, lines of the geometry are represented by geodesic s on the pseudosphere and theorems ..., Ferdinand Minding Minding already considered geodesic triangles on the pseudosphere and remarked ... . It is often stated that this proof was incomplete due to the singularities of the pseudosphere, which ... must have been well aware of this difficulty, which is also manifested by the fact that the pseudosphere ... metric metric on the pseudosphere can be transferred to the unit disk and that the singularity theory singularity of the pseudosphere corresponds to a horocycle on the non Euclidean plane. On the other ... more details
This is a list of surface s , by Wikipedia page. See also List of algebraic surfaces , List of curves , Riemann surface . Minimal surface s Costa surface Catenoid Helicoid Riemann s surface Saddle tower Gyroid Catalan surface Enneper surface orientability Non orientable surfaces Klein bottle Real projective plane Cross cap Roman surface Boy s surface Quadric s Sphere Spheroid Oblate spheroid Cone geometry Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets hyperbolic paraboloid a ruled surface Paraboloid Pseudospherical surfaces Dini s surface Pseudosphere Algebraic surface s See the list of algebraic surfaces . Cayley cubic Barth sextic Clebsch cubic Monkey saddle saddle like surface for 3 legs. Torus Dupin cyclide inversion of a torus Whitney umbrella Miscellaneous surfaces right conoid a Ruled surface Category Mathematics related lists Surfaces Category Surfaces pt Anexo Lista de superf cies sl Seznam ploskev ... more details
Models of non Euclidean geometry are mathematical model s of geometries in which are non Euclidean geometry non Euclidean in the sense that it is not the case that exactly one line can be drawn parallel lines parallel to a given line l through a point that is not on l . In hyperbolic geometric models, by contrast, there are infinity infinitely many lines through A parallel to l , and in elliptic geometric models, parallel lines do not exist. See the entries on hyperbolic geometry and elliptic geometry for more information. Euclidean geometry is modelled by our notion of a flat plane mathematics plane . The simplest model for elliptic geometry is a sphere, where lines are great circle s such as the equator or the meridian geography meridian s on a globe , and points opposite each other are identified considered to be the same . The pseudosphere has the appropriate curvature to model hyperbolic geometry. See also Projective geometry Spherical geometry Taxicab geometry Riemannian geometry References Ian Stewart. Flatterland . Perseus Publishing ISBN 0 7382 0675 X softcover, 2001 Marvin Jay Greenberg. Euclidean and non Euclidean geometries Development and history . Publisher W H Freeman 1993. ISBN 0 7167 2446 4. External links http www groups.dcs.st and.ac.uk history HistTopics Non Euclidean geometry.html MacTutor Archive article on non Euclidean geometry Category Classical geometry es Modelos de geometr a no euclidiana ... more details
Unreferenced date December 2009 Seealso Space form In mathematics , constant curvature in differential geometry is a concept most commonly applied to surface s. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points. The circle has constant curvature, also, in a natural but different sense. The standard surface geometries of constant curvature are elliptic geometry or spherical geometry which has positive curvature , Euclidean geometry which has zero curvature, and hyperbolic geometry pseudosphere geometry which has negative curvature . Since Riemann surface s can be taken to have constant curvature, there is a large supply of other examples, for negative curvature. For higher dimensional manifold s, constant curvature is usually taken to mean constant sectional curvature , and a complete manifold of this kind is called a space form . As in the case of surfaces, there are three types of geometries elliptic, flat, or hyperbolic according to whether the curvature is positive, zero, or negative. The universal cover of a manifold of constant sectional curvature is one of the model spaces sphere, Euclidean space, hyperbolic space , and the study of space forms is thus generalized crystallography. Spherical manifold Flat manifold Hyperbolic manifold Seealso Curvature of Riemannian manifolds DEFAULTSORT Constant Curvature Category Differential geometry Category Riemannian geometry nl Constante kromming zh ... more details
Infobox scientist name PAGENAME image Ferdinand Minding.jpg image size 200 px caption birth date Dec 30, 1805 Jan 11, 1806 birth place Kalisz death date May 1, 1885 May 13, 1885 death place Dorpat residence citizenship Russia Russian nationality ethnicity field Mathematics work institutions University of Berlin , University of Dorpat alma mater University of Halle doctoral advisor doctoral students known for invariance of geodesic curvature , geodesic s on pseudosphere , bending of surfaces author abbrev bot author abbrev zoo influences Gauss influenced Karl Peterson prizes Demidov Prize of St Peterburg Academy of Sciences 1861 religion footnotes Birth and death dates are given according to Neue Deutsche Biographie in Julian and Gregorian calendar signature Ferdinand Minding lang ru OldStyleDateDY January 11 1806 December 30 1805 &ndash OldStyleDate May 13 1885 May 1 was a German people German Russian people Russian mathematician known for his contributions to differential geometry . He continued the work of Gauss concerning differential geometry of surfaces , especially its intrinsic aspects. Minding considered questions of bending of surfaces and proved the invariance of geodesic curvature . He studied ruled surface s, developable surface s and surface of revolution surfaces of revolution and determined geodesics on the pseudosphere . Minding s results on the geometry of geodesic triangles on a surface of constant curvature 1840 anticipated Eugenio Beltrami Beltrami s approach to the foundations of non Euclidean geometry 1868 . Minding was largely self taught in mathematics. He attended lectures in the University of Halle and eventually graduated with a thesis De valore intergralium duplicium quam proxime inveniendo 1829 . Minding worked as a teacher in Elberfeld and as a university lecturer in Berlin. His work on statics drew attention of Alexander von Humboldt . However, his 1842 bid for election to Berlin Academy , supported by Diri ... more details
Image Emanoil Bacalogu.jpg thumb right Emanoil Bacaloglu Emanoil Bacaloglu IPA ro emano il baka lo lu 11 April 1830 &ndash 30 August 1891 was a Wallachia n and Romania n mathematician , physicist and chemist and a scubadiver, medic, shoe salesman, a vegetarian,policeman and fireman. Born in Bucharest and of Greeks of Romania Greek origin, he studied physics and mathematics in Paris and Leipzig , later becoming a professor at the University of Bucharest and, in 1879, a member of the Romanian Academy . Considered to be the founder of many scientific and technological fields in Romania and aiding in the creation of the Romanian Athenaeum , Bacaloglu was also an accomplished scientist. He helped create Romanian language terminology in his fields and was one of the principal founders of the Society of Physical Sciences in 1890. He was also a participant in the 1848 Wallachian revolution . He is known for the Bacaloglu pseudosphere . This is a surface of revolution for which the Bacaloglu curvature is constant. Main works Elemente de fizic , 2 nd edition, Bucure ti, 1888 Elemente de algebr , 2 nd edition, Bucure ti, 1870 References Florica C mpan, La pseudosph re de Bacaloglu , Acad. Roum. Bull. Sect. Sci. 24 1943 , 96&ndash 105. MathSciNet id 0024191 External links http www.ici.ro romania en stiinta bacaloglu.html Short bio http www.physics.pub.ro NOU Scurt 20istoric.htm Short history , at the Polytechnic University of Bucharest Persondata Metadata see Wikipedia Persondata . NAME Bacaloglu, Emanoil ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 11 April 1830 PLACE OF BIRTH DATE OF DEATH 30 August 1891 PLACE OF DEATH DEFAULTSORT Bacaloglu, Emanoil Category Romanian mathematicians Category 19th century mathematicians Category Romanian physicists Category Romanian chemists Category Romanian Academy Category People of the Revolutions of 1848 Category University of Bucharest faculty Category People from Bucharest Category 1830 births Category 1891 deaths Romania scientist ... more details
File Dini s Surface.svg thumb 350px right Dini s surface with 0 u 4 &pi and 0.01 v 1 and constants a 1.0 and b 0.2. In geometry , Dini s surface is a surface with constant negative curvature that can be created by twisting a pseudosphere . ref cite web url http mathworld.wolfram.com DinisSurface.html title Wolfram Mathworld Dini s Surface accessdate 2009 11 12 ref It is named after Ulisse Dini ref cite web url http www gap.dcs.st and.ac.uk history Biographies Dini.html title Ulisse Dini Biography publisher School of Mathematics and Statistics, University of St Andrews, Scotland year 2000 author J J O Connor and E F Robertson accessdate 209 11 14 ref and described by the following parametric equation s ref cite web url http knol.google.com k suresh emre dini s surface geometry 35vsnxisjn2mw 26 title Knol Dini s Surface geometry accessdate 2009 11 12 ref math x a cos left u right sin left v right math math y a sin left u right sin left v right math math z a left cos left v right log left tan left frac v 2 right right right bu math Another description is a helicoid constructed from the tractrix . ref cite book title B cklund and Darboux transformations geometry and modern applications in Soliton Theory last Rogers and Schief publisher Cambridge University Press pages 35 36 year 2002 ref Uses Renditions of Dini s surface have appeared on the covers of Western Kentucky University s Graduate Study in Mathematics , Alfred Gray mathematician Gray s Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition , and volume 2, number 3 of La Gaceta de la Real Sociedad Matem tica Espa ola . See also Dini derivative Dini test Dini s theorem List of surfaces References reflist Category Surfaces it Superficie di Dini tr Dini y zeyi ... more details
Unreferenced date February 2007 Cyrillic alphabet navbox Heading Cyrillic letter br Ye with grave Image File Cyrillic letter Ye with grave uppercase and lowercase.svg 120px uuc 0400 ulc 0450 Ye with grave italics span style font family times, Times New Roman , serif font size larger span is a regular combination of Cyrillic script Cyrillic Ye Cyrillic letter Ye and grave accent . This combination is not a separate letter of the alphabet of any language, and Wikipedia has an individual article about this combination primarily because it has individual position in certain computer encodings such as Unicode. Usage Ye with grave represents a Stress linguistics stressed variant of the Ye Cyrillic Cyrillic letter Ye . Most regularly it is used in Macedonian language Macedonian to prevent ambiguity in certain cases lang mk , Lord s Prayer And do not lead us into temptation, but deliver us from evil , or lang mk All that you ll write can be used literally it can use itself against you , etc. Ye with grave can also be found in accented Bulgarian language Bulgarian , Serbian language Serbian or Church Slavonic texts, as well as in older 19th century or earlier Russian books. Recently, Russian language Russian stressed vowels are typically marked with the acute accent instead of the grave accent , and the role of grave accent is limited to the secondary stress mark in certain dictionaries acute accent shows the main stress pseudosphere . Computing codes class wikitable style text align right align center align right character colspan 2 colspan 2 align center align right Unicode name colspan 2 small CYRILLIC CAPITAL LETTER br IE WITH GRAVE small colspan 2 small CYRILLIC SMALL LETTER br IE WITH GRAVE small align left character encoding decimal hex decimal hex align left Unicode 1024 0400 1104 0450 align left UTF 8 208 128 D0 ... more details
of the revolution surface of it around its asymptote the pseudosphere . Studied by Eugenio Beltrami Beltrami in 1868, as a surface of constant negative Gaussian curvature , the pseudosphere is a local ... frac y a math . The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere ... more details
A Picard horn , also called the Picard topology or Picard model , is a theoretical model for the shape of the Universe . It is a horn topology, meaning it has hyperbolic geometry the term horn is due to pseudosphere models of hyperbolic space . The term was coined by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper Hyperbolic Universes with a Horned Topology and the CMB Anisotropy . ref name Aurich0403597 cite journal last Aurich first Ralf coauthors Lustig, S., Steiner, F., Then, H. title Hyperbolic Universes with a Horned Topology and the CMB Anisotropy journal Classical and Quantum Gravity volume 21 pages 4901 4926 publisher Institute of Physics year 2004 url http iopscience.iop.org 0264 9381 21 21 010 doi 10.1088 0264 9381 21 21 010 accessdate 2011 08 24 archiveurl http arXiv.org abs astro ph 0403597 archivedate 2004 10 14 arxiv astro ph 0403597 bibcode 2004CQGra..21.4901A ref The space in question is the quotient of the Poincar half plane model upper half plane model of hyperbolic 3 space by the group math operatorname PSL 2 mathbf Z i math , which was first described by mile Picard ref name EmilePicard http www.academie sciences.fr activite archive dossiers Picard Picard oeuvre.htm ref in 1884. ref name picard1884 cite journal author Emile Picard language French language French title Sur un groupe de transformations des points de l espace situ s du m me c t d un plan journal Bulletin de la Soci t Math matique de France volume 12 pages 43 37 date 1884 03 07 url http www.numdam.org item?id BSMF 1884 12 43 0 accessdate 2011 08 24 ref A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Fritz Grunewald and Wolfgang Huntebrinker, http projecteuclid.org euclid.em 1047591148 A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp , Experiment. Math. Volume 5, Issue 1 1996 , 57 80. The same source calculates the first 80 eigenvalues of the Laplacian ... more details
File Mobius Strip.jpg thumb Large mobius strip with traveling arrow. Mathematica A World of Numbers and Beyond is an interactive exhibition originally at the California Museum of Science and Industry . Duplicates have since been made, and they as well as the original have been moved to other institutes. History Image Multiplication Machine.jpg thumb left Multiplication machine in the exhibit. File The Pseudosphere.jpg thumb Pseudosphere model on display. In March, 1961 a new science wing at the California Museum of Science and Industry ref name CMS Called the California Science Center since 1998. ref in Los Angeles opened. IBM had been asked by the Museum to make a contribution IBM in turn asked the famous California designer team of Charles Eames and his wife Ray Eames to come up with a good proposal. The result was that the Eames Office was commissioned by IBM to design an interactive exhibition called Mathematica a world of numbers...and beyond . ref name rights The physical component of the exhibit was owned by the Museum, it was financially supported by IBM, and the Eames Office retained the artistic property rights. ref This was the first of many exhibitions designed by the Eames Office. This convert 3000 sqft m2 sing on exhibition stayed at the Museum until January 1998, making it the longest running of any corporate sponsored museum exhibition. http www.designboom.com eng funclub mathematica.html Furthermore, it is the only one of the dozens of exhibitions designed by the Eames Office that is still extant. This original Mathematica Exhibition is now owned by and on display at the New York Hall of Science . http www.nyscience.org exhibitions explore exhibitions 38054 Duplicates In November, 1961 a duplicate was made for Chicago s Museum of Science and Industry Chicago Museum of Science and Industry http www.msichicago.org where it stayed until late 1980. From there it moved to the Museum of Science, Boston http www.mos.org exhibitdevelopment mosExhibits.html ... more details
In mathematics , B cklund transforms or B cklund transformations called after the Swedish mathematician Albert Victor B cklund relate partial differential equation s and their solutions. They are an important tool in soliton theory and integrable system s. A B cklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a B cklund transformation of the other. A B cklund transform which relates solutions of the same equation is called an invariant B cklund transform or auto B cklund transform . If such a transform can be found, much can be deduced about the solutions of the equation especially if the B cklund transform contains a parameter. However, no systematic way of finding B cklund transforms is known. History File Pseudosphere.png thumb B cklund transforms originated as transformations of pseudosphere s in the 1880s. B cklund transforms have their origins in differential geometry the first nontrivial example is the transformation of pseudospherical surface s introduced by Luigi Bianchi L. Bianchi and Albert Victor B cklund A.V. B cklund in the 1880s. This is a geometrical construction of a new pseudospherical surface from an initial such surface using a solution of a linear differential equation . Pseudospherical surfaces can be described as solutions of the sine Gordon equation , and hence the B cklund transformation of surfaces can be viewed as a transformation of solutions of the sine Gordon equation. The Cauchy Riemann equations see Cauchy Riemann equations The prototypical example of a B cklund transform is the Cauchy Riemann equations Cauchy Riemann system math u x v y, quad u y v x, , math which relates the real and imaginary parts u and v of a holomorphic function . This first order system of partial differential equations has ... more details
The Institute For Figuring IFF is an organization based in Los Angeles, California that promotes the public understanding of the poetry poetic and aesthetics aesthetic dimensions of science , mathematics and the technical arts. The Institute hosts public lecture s and Art exhibition exhibition s, publishes books and maintains a website . Overview Image Orange anemonie.jpg thumb 200px right Crocheted hyperbolic pseudosphere from the IFF Collection. Image MS Jeannine.jpg thumb 200px right Dr Jeannine Mosely and the Business Card Menger Sponge. Image Crochet hyperbolic kelp.jpg 200px thumb The IFF s collection of crocheted hyperbolic planes, in imitation of a coral reef. Image Hyperkelp.jpg thumb 200px right IFF hyperbolic crochet cacti and kelp at the LA County Fair, 2006. According to the organization s website, the Institute For Figuring is dedicated to enhancing the public understanding of figures and figuring techniques. From the physics of snowflakes and the hyperbolic geometry of sea slugs, to the mathematics of paper folding and graphical models of the human mind, the Institute takes as its purview a complex ecology of figuring. Since its founding in 2003 by Margaret Wertheim and Christine Wertheim the IFF has staged public lectures in Los Angeles and New York City New York on subjects such as tiling patterns, hyperbolic space, early computational devices, and tensegrity structures. The Wales Welsh writer Merrily Harpur has written that the duty of artists everywhere is to enchant the conceptual landscape. The IFF was founded on the principle that science, mathematics and other techno logical pursuits may also achieve this goal. In spring 2006 a lecture series at Telic Arts Exchange in Los Angeles, entitled The Insect Trilogy, presented leading zoologists talking about how flies fly, how spiders see, and the ecology of a termite s gut. In New York, the Institute co hosts events with Cabinet magazine Cabinet magazine, an international arts and culture quarterly. ... more details
1 0 1 1 the subcase of type VI sub a sub with math a 1 math IV 1 0 0 1 V 1 0 0 0 has a hyper pseudosphere ... a math 0 1 1 has a hyper pseudosphere as a special case VIII 0 1 1 1 IX 0 1 1 1 has a hypersphere ... more details
by an isometric embedding in a flat space of one higher dimension as the sphere and pseudosphere ... space , effectively unrolling the embedding. A similar situation occurs with the pseudosphere, which ... more details
The sine Gordon equation is a nonlinear hyperbolic partial differential equation in 1 1 dimensions involving the d Alembert operator and the sine function sine of the unknown function. It was originally considered in the nineteenth century in the course of study of pseudosphere surfaces of constant negative curvature . This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions. Origin of the equation and its name There are two equivalent forms of the sine Gordon equation. In the real number real space time coordinates , denoted x , t , the equation reads math , varphi tt varphi xx sin varphi 0. math Passing to the light cone coordinates u , v , akin to asymptotic coordinates where math u frac x t 2, quad v frac x t 2, math the equation takes the form math varphi uv sin varphi. , math This is the original form of the sine Gordon equation, as it was considered in the nineteenth century in the course of investigation of differential geometry of surfaces surfaces of constant Gaussian curvature K &minus 1, also called pseudospherical surface s. Choose a coordinate system for such a surface in which the coordinate mesh u constant, v constant is given by the asymptotic curve asymptotic line s parameterized with respect to the arc length. The first fundamental form of the surface in these coordinates has a special form math ds 2 du 2 2 cos varphi ,du , dv dv 2, , math where expresses the angle between the asymptotic lines, and for the second fundamental form , L N 0. Then the Codazzi Mainardi equation expressing a compatibility condition between the first and second fundamental forms results in the sine Gordon equation. The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Luigi Bianchi Bianchi and Albert Victor B cklund B cklund led to the discovery of B cklund transformation s. The name sine ... more details
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