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Hyperbolic geometry

Index Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry. [1]

175 relations: Abner of Burgos, Absolute geometry, Absolute scale, Adrien-Marie Legendre, American Mathematical Monthly, American Mathematical Society, Analytic philosophy, Angle, Angle of parallelism, Angular defect, Anti-de Sitter space, Apeirogon, Arthur Cayley, Atlas (topology), Axiom, Baruch Spinoza, Beltrami–Klein model, Bisection, Book of Optics, Bookseller/Diagram Prize for Oddest Title of the Year, Carl Friedrich Gauss, Cartesian coordinate system, Cayley–Klein metric, Chord (geometry), Circle Limit III, Circumscribed circle, Conformal map, Conic section, Coordinate system, Course of Theoretical Physics, Crochet, Cross-ratio, Curvature, Daina Taimina, De Sitter space, Degrees of freedom, E (mathematical constant), Earth's orbit, Elliptic geometry, Encyclopedia of the History of Arabic Science, Euclid, Euclid's Elements, Euclidean geometry, Eugenio Beltrami, Exponential growth, Face (geometry), Felix Klein, Franz Taurinus, Galilean invariance, Galilean transformation, ..., Gaussian curvature, Geodesic, Geodesic curvature, Geometric transformation, Geometrization conjecture, Geometry, Gersonides, Giovanni Girolamo Saccheri, Gyrovector space, Harold Scott MacDonald Coxeter, Helaman Ferguson, Henri Poincaré, Hilbert's theorem (differential geometry), Hjelmslev transformation, Homothetic transformation, Horocycle, Hyperbola, Hyperbolic 3-manifold, Hyperbolic function, Hyperbolic geometry, Hyperbolic manifold, Hyperbolic set, Hyperbolic space, Hyperbolic tree, Hyperbolic triangle, Hyperboloid, Hyperboloid model, Hypercycle (geometry), HyperRogue, Ibn al-Haytham, Ideal point, Ideal triangle, Identity function, If and only if, Immanuel Kant, Immersion (mathematics), Incircle and excircles of a triangle, Isometry, Jakob Nielsen (mathematician), János Bolyai, Jeffrey Weeks (mathematician), Johann Heinrich Lambert, John Milnor, John Wallis, Kleinian group, Lambert quadrilateral, Limiting parallel, List of geometers, Logic, M. C. Escher, Margin of error, Mathematical model, Mathematics, Möbius transformation, Metric (mathematics), Metric space, Minkowski space, Motion (geometry), Nasir al-Din al-Tusi, Nikolai Lobachevsky, Non-Euclidean geometry, Normal (geometry), Omar Khayyam, Orthogonal group, Orthogonality, Orthographic projection, Parallax, Parallel postulate, Parsec, Perpendicular, Philosophy, Plane (geometry), Playfair's axiom, Poincaré disk model, Poincaré half-plane model, Poincaré metric, Point reflection, Positive real numbers, Post-Soviet states, Proclus, Projection (mathematics), Projective geometry, Proof by contradiction, Proper time, Pseudosphere, Quadric, Quadrilateral, Radian, Rapidity, Reflection (mathematics), Regular polygon, Riemann sphere, Rigour, Roguelike, Routledge, Saccheri quadrilateral, Saddle point, Schwarz triangle, Shape of the universe, Sirius, Spacetime, Special relativity, Sphere-world, Spherical geometry, Stack Exchange, Stereographic projection, Symmetric space, Systolic geometry, Tangent, The Daily Telegraph, The Feynman Lectures on Physics, Thomas Hobbes, Thought experiment, Triangle group, Truncated order-7 triangular tiling, Two-dimensional space, Ultraparallel theorem, Uniform tilings in hyperbolic plane, Unit circle, Unit sphere, University of Glasgow, Vitello, Werner Fenchel, Wilhelm Killing, William Thurston. Expand index (125 more) »

Abner of Burgos

Abner of Burgos (c. 1270 – c. 1347, or a little later) was a Jewish philosopher, a convert to Christianity and polemical writer against his former religion.

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Absolute geometry

Absolute geometry is a geometry based on an axiom system for Euclidean geometry with the parallel postulate removed and none of its alternatives used in place of it.

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Absolute scale

An absolute scale is a system of measurement that begins at a minimum, or zero point, and progresses in only one direction.

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Adrien-Marie Legendre

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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Analytic philosophy

Analytic philosophy (sometimes analytical philosophy) is a style of philosophy that became dominant in the Western world at the beginning of the 20th century.

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In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

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Angle of parallelism

In hyperbolic geometry, the angle of parallelism \Pi(a), is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel sides.

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Angular defect

In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would.

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Anti-de Sitter space

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature.

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In geometry, an apeirogon (from the Greek word ἄπειρος apeiros, "infinite, boundless" and γωνία gonia, "angle") is a generalized polygon with a countably infinite number of sides.

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Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

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Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

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An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

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Baruch Spinoza

Baruch Spinoza (born Benedito de Espinosa,; 24 November 1632 – 21 February 1677, later Benedict de Spinoza) was a Dutch philosopher of Sephardi/Portuguese origin.

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Beltrami–Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

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In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector.

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Book of Optics

The Book of Optics (Kitāb al-Manāẓir; Latin: De Aspectibus or Perspectiva; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen (965– c. 1040 AD).

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Bookseller/Diagram Prize for Oddest Title of the Year

The Bookseller/Diagram Prize for Oddest Title of the Year, originally known as the Diagram Group Prize for the Oddest Title at the Frankfurt Book Fair, commonly known as the Diagram Prize for short, is a humorous literary award that is given annually to a book with an unusual title.

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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

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Cayley–Klein metric

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space is defined using a cross-ratio.

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Chord (geometry)

A chord of a circle is a straight line segment whose endpoints both lie on the circle.

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Circle Limit III

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came".

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Circumscribed circle

In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon.

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Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

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Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Course of Theoretical Physics

The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s.

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Not to be confused with Crotchet, the common name for a Quarter note in music. Crochet is a process of creating fabric by interlocking loops of yarn, thread, or strands of other materials using a crochet hook.

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In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line.

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In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

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Daina Taimina

Daina Taimina (Taimiņa; born August 19, 1954) is a Latvian mathematician, currently Adjunct Associate Professor at Cornell University, known for crocheting objects to illustrate hyperbolic space.

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De Sitter space

In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary Euclidean space.

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Degrees of freedom

In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently.

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E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

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Earth's orbit

Earth's orbit is the trajectory along which Earth travels around the Sun.

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Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

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Encyclopedia of the History of Arabic Science

The Encyclopedia of the History of Arabic Science is a three-volume encyclopedia covering the history of Arabic contributions to science, mathematics and technology which had a marked influence on the Middle Ages in Europe.

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Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

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Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

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Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

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Eugenio Beltrami

Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics.

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Exponential growth

Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function is proportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent.

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Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

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Felix Klein

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.

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Franz Taurinus

Franz Adolph Taurinus (15 November 1794 – 13 February 1874) was a German mathematician who is known for his work on non-Euclidean geometry.

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Galilean invariance

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames.

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Galilean transformation

In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics.

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Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere.

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In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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Geodesic curvature

In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic.

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Geometric transformation

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set.

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Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.

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Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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Levi ben Gershon (1288–1344), better known by his Graecized name as Gersonides or by his Latinized name Magister Leo Hebraeus the abbreviation of first letters as RaLBaG, was a medieval French Jewish philosopher, Talmudist, mathematician, physician and astronomer/astrologer.

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Giovanni Girolamo Saccheri

Giovanni Girolamo Saccheri (5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician.

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Gyrovector space

A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Helaman Ferguson

Helaman Rolfe Pratt Ferguson (born 1940 in Salt Lake City, Utah) is an American sculptor and a digital artist, specifically an algorist.

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Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

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Hilbert's theorem (differential geometry)

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative gaussian curvature K immersed in \mathbb^.

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Hjelmslev transformation

In mathematics, the Hjelmslev transformation is an effective method for mapping an entire hyperbolic plane into a circle with a finite radius.

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Homothetic transformation

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if) or reverse (if) the direction of all vectors.

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In hyperbolic geometry, a horocycle (ὅριον + κύκλος — border + circle, sometimes called an oricycle, oricircle, or limit circle) is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction.

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In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set.

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Hyperbolic 3-manifold

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1.

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Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.

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Hyperbolic set

In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under f, with respect to some Riemannian metric on M. An analogous definition applies to the case of flows.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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Hyperbolic tree

A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry.

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Hyperbolic triangle

In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane.

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In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.

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Hyperboloid model

In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model (after Hermann Minkowski and Hendrik Lorentz), is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space and m-planes are represented by the intersections of the (m+1)-planes in Minkowski space with S+.

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Hypercycle (geometry)

In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis).

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HyperRogue is an independent video game developed by Zeno Rogue.

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Ibn al-Haytham

Hasan Ibn al-Haytham (Latinized Alhazen; full name أبو علي، الحسن بن الحسن بن الهيثم) was an Arab mathematician, astronomer, and physicist of the Islamic Golden Age.

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Ideal point

In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space.

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Ideal triangle

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points.

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Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Immanuel Kant

Immanuel Kant (22 April 1724 – 12 February 1804) was a German philosopher who is a central figure in modern philosophy.

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Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.

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Incircle and excircles of a triangle

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides.

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In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

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Jakob Nielsen (mathematician)

Jakob Nielsen (15 October 1890 in Mjels, Als – 3 August 1959 in Helsingør) was a Danish mathematician known for his work on automorphisms of surfaces.

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János Bolyai

János Bolyai (15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, one of the founders of non-Euclidean geometry — a geometry that differs from Euclidean geometry in its definition of parallel lines.

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Jeffrey Weeks (mathematician)

Jeffrey Renwick Weeks (born December 10, 1956) is an American mathematician, a geometric topologist and cosmologist.

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Johann Heinrich Lambert

Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.

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John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems.

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John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

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Kleinian group

In mathematics, a Kleinian group is a discrete subgroup of PSL(2, '''C''').

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Lambert quadrilateral

In geometry, a Lambert quadrilateral, named after Johann Heinrich Lambert, is a quadrilateral in which three of its angles are right angles.

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Limiting parallel

In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R).

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List of geometers

A geometer is a mathematician whose area of study is geometry.

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Logic (from the logikḗ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.

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M. C. Escher

Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically-inspired woodcuts, lithographs, and mezzotints.

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Margin of error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results.

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Mathematical model

A mathematical model is a description of a system using mathematical concepts and language.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

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Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

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Motion (geometry)

In geometry, a motion is an isometry of a metric space.

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Nasir al-Din al-Tusi

Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (محمد بن محمد بن حسن طوسی‎ 18 February 1201 – 26 June 1274), better known as Nasir al-Din Tusi (نصیر الدین طوسی; or simply Tusi in the West), was a Persian polymath, architect, philosopher, physician, scientist, and theologian.

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Nikolai Lobachevsky

Nikolai Ivanovich Lobachevsky (a; –) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry and also his fundamental study on Dirichlet integrals known as Lobachevsky integral formula.

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Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

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Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

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Omar Khayyam

Omar Khayyam (عمر خیّام; 18 May 1048 – 4 December 1131) was a Persian mathematician, astronomer, and poet.

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Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

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In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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Orthographic projection

Orthographic projection (sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions.

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Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.

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Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry.

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The parsec (symbol: pc) is a unit of length used to measure large distances to astronomical objects outside the Solar System.

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In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).

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Philosophy (from Greek φιλοσοφία, philosophia, literally "love of wisdom") is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.

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Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

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Playfair's axiom

In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

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Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

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Poincaré half-plane model

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

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Poincaré metric

In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature.

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Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

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Positive real numbers

In mathematics, the set of positive real numbers, \mathbb_.

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Post-Soviet states

The post-Soviet states, also collectively known as the former Soviet Union (FSU) or former Soviet Republics, are the states that emerged and re-emerged from the Union of Soviet Socialist Republics in its breakup in 1991, with Russia internationally recognised as the successor state to the Soviet Union after the Cold War.

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Proclus Lycaeus (8 February 412 – 17 April 485 AD), called the Successor (Greek Πρόκλος ὁ Διάδοχος, Próklos ho Diádokhos), was a Greek Neoplatonist philosopher, one of the last major classical philosophers (see Damascius).

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Projection (mathematics)

In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition (or, in other words, which is idempotent).

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Projective geometry

Projective geometry is a topic in mathematics.

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Proof by contradiction

In logic, proof by contradiction is a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition.

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Proper time

In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line.

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In geometry, the term pseudosphere is used to describe various surfaces with constant negative Gaussian curvature.

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In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

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In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners.

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The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

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In relativity, rapidity is commonly used as a measure for relativistic velocity.

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Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

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Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

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Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

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Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

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Roguelike is a subgenre of role-playing video game characterized by a dungeon crawl through procedurally generated levels, turn-based gameplay, tile-based graphics, and permanent death of the player character.

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Routledge is a British multinational publisher.

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Saccheri quadrilateral

A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base.

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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) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.

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Schwarz triangle

In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges.

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Shape of the universe

The shape of the universe is the local and global geometry of the universe.

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Sirius (a romanization of Greek Σείριος, Seirios,."glowing" or "scorching") is a star system and the brightest star in the Earth's night sky.

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In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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The idea of a sphere-world was constructed by Henri Poincaré who, while pursuing his argument for conventionalism (see philosophy of space and time), offered a thought experiment about a sphere with strange properties.

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Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere.

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Stack Exchange

Stack Exchange is a network of question-and-answer (Q&A) websites on topics in varied fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process.

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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Symmetric space

In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.

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Systolic geometry

In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations.

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In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

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The Daily Telegraph

The Daily Telegraph, commonly referred to simply as The Telegraph, is a national British daily broadsheet newspaper published in London by Telegraph Media Group and distributed across the United Kingdom and internationally.

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The Feynman Lectures on Physics

The Feynman Lectures on Physics is a physics textbook based on some lectures by Richard P. Feynman, a Nobel laureate who has sometimes been called "The Great Explainer".

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Thomas Hobbes

Thomas Hobbes (5 April 1588 – 4 December 1679), in some older texts Thomas Hobbes of Malmesbury, was an English philosopher who is considered one of the founders of modern political philosophy.

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Thought experiment

A thought experiment (Gedankenexperiment, Gedanken-Experiment or Gedankenerfahrung) considers some hypothesis, theory, or principle for the purpose of thinking through its consequences.

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Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle.

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Truncated order-7 triangular tiling

In geometry, the Order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane.

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Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

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Ultraparallel theorem

In hyperbolic geometry, the ultraparallel theorem states that every pair of ultraparallel lines (lines that are not intersecting and not limiting parallel) has a unique common perpendicular hyperbolic line.

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Uniform tilings in hyperbolic plane

In hyperbolic geometry, a uniform (regular, quasiregular or semiregular) hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).

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Unit circle

In mathematics, a unit circle is a circle with a radius of one.

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Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.

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University of Glasgow

The University of Glasgow (Oilthigh Ghlaschu; Universitas Glasguensis; abbreviated as Glas. in post-nominals) is the fourth-oldest university in the English-speaking world and one of Scotland's four ancient universities.

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Witelo (also Erazmus Ciołek Witelo; Witelon; Vitellio; Vitello; Vitello Thuringopolonis; Vitulon; Erazm Ciołek); born ca.

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Werner Fenchel

Moritz Werner Fenchel (3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory.

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Wilhelm Killing

Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.

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William Thurston

William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.

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Analytic hyperbolic geometry, Bolyai geometry, Bolyai surface, Bolyai-Lobachevskian geometry, Bolyai-Lobachevskian surface, Gans model, Gauss-Bolyai-Lobachevsky space, Gauss–Bolyai–Lobachevsky space, Hemisphere model, Hyperbolic Geometry, Hyperbolic plane, Hyperbolic plane (geometry), Hyperbolic surface, Knit theory, Lobachevski plane, Lobachevskian, Lobachevskian geometry, Lobachevskian or hyperbolic geometry, Lobachevskii geometry, Lobachevskii space, Lobachevsky geometry, Lobachevsky plane, Lobachevsky-Bolyai-Gauss Geometry, Models of the hyperbolic plane, Ultraparallel, Ultraparallel line, Universal hyperbolic geometry.


[1] https://en.wikipedia.org/wiki/Hyperbolic_geometry

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