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

Look up hyperbolic in Wiktionary, the free dictionary. Hyperbolic may refer to: of or pertaining to a hyperbola, a type of smooth curve lying in a plane...

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just...

common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse...

Hyperbole (/haɪˈpɜːrbəli/ hy-PUR-bə-lee; adj. hyperbolic /ˌhaɪpərˈbɒlɪk/ HY-pur-BOL-ick) is the use of exaggeration as a rhetorical device or figure of...

Hyperbolic is the sixth studio album by Australian electronic music trio Pnau, released on 22 March 2024 through Etcetc Music and later reissued through...

In mathematics, relatively hyperbolic groups form an important class of groups of interest for geometric group theory. The main purpose in their study...

In economics, hyperbolic discounting is a time-inconsistent model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis...

In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane...

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature...

systems theory, a subset Λ of a smooth manifold M is said to be hyperbolic or to have a hyperbolic structure with respect to a smooth map f if its tangent bundle...

an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space. This...

In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit (from Newtonian theory: hyperbola shape) is the trajectory of any...

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number...

precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...

This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank....

In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three...

forms associated with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries...

A hyperbolic asteroid is any sort of asteroid or non-cometary astronomical object observed to have an orbit not bound to the Sun and will have an orbital...

plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines...

A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described...

Circular orbit: e = 0 Elliptic orbit: 0 < e < 1 Parabolic trajectory: e = 1 Hyperbolic trajectory: e > 1 The eccentricity e is given by e = 1 +   2   E   L 2...

In mathematics, a hyperbolic partial differential equation of order n {\displaystyle n} is a partial differential equation (PDE) that, roughly speaking...

{cl} z=\operatorname {cd} (z;-1)} . Similarly, the hyperbolic lemniscate sine slh and hyperbolic lemniscate cosine clh have a square period lattice with...

hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic...

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in...

finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1 / x {\displaystyle 1/x}...

complete set of hyperbolic uniform honeycombs. More unsolved problems in mathematics In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform...

A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a)...

given a pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described by Apollonius...

extensively. Important subbranches are Hyperbolic dynamics, non uniform hyperbolicity and partial hyperbolicity. Assume a two dimensional manifold, and...

In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using...

Hyperbolic navigation is a class of radio navigation systems in which a navigation receiver instrument is used to determine location based on the difference...

global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It is called hyperbolic in analogy...

In mathematics, hyperbolic trigonometry can mean: The study of hyperbolic triangles in hyperbolic geometry (traditional trigonometry is the study of triangles...

A hyperbolic paraboloid structure (also called a hypar) is a load-bearing structure with the shape of a truncated region of an infinite hyperbolic paraboloid...

In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form q = a + b i + c...

In mathematics, the complex hyperbolic space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds...

topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric...

In mathematics, a hyperbolic point is a certain kind of point, one of: A point in a hyperbolic geometry A point of negative Gaussian curvature on a smooth...

The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola...

In mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called...

In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous...

are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution...

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals...

Hyperbolic theory may refer to: Hyperbolic geometry The theory of hyperbolic partial differential equations This disambiguation page lists mathematics...

is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations...

In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p...

In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic...

In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space...

hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn...

mathematics at Cornell University, known for developing a way of modeling hyperbolic geometry with crocheted objects. Taimiņa received all of her formal education...

boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually...

cone Hyperbolic cylinder Hyperbolic paraboloid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Hyperbolic cylinder Hyperbolic paraboloid...

In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane { ( x , y )   :   x > 0 ,   y > 0   } = Q {\displaystyle...

The Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta...

knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume...

Husemoller for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited. The affine hyperbolic plane was described...

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space....

systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the...

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

algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle...

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of...

way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using...

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar...

In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space...

Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation...

equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as I α...

model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines...

manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the...

subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use...

The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution...

professor of mathematics at Stanford University. Her research focused on hyperbolic geometry, dynamical systems, complex analysis, and topology. In 2014,...

{PSL} _{2}(\mathbb {Z} )} . They, and the hyperbolic surface associated to their action on the hyperbolic plane, often exhibit particularly regular behaviour...

Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the...

Contributed idea that a very large class of closed 3-manifolds carry a hyperbolic structure." Shing-Tung Yau Institute for Advanced Study, US Tsinghua University...

simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials, and...

special relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper...

special relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper...

each other along that side. Hyperbolic polygons are the analogues of Euclidean polygons in hyperbolic geometry. A hyperbolic n-gon is an area bounded by...

four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around...

orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds. In physics, the identity...

function of the logistic distribution is also a scaled version of the hyperbolic tangent. F ( x ; μ , s ) = 1 1 + e − ( x − μ ) / s = 1 2 + 1 2 tanh ⁡...

geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter...

Crocheting Adventures with Hyperbolic Planes is a book on crochet and hyperbolic geometry by Daina Taimiņa. It was published in 2009 by A K Peters, with...

made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for...

A saddle roof is a hyperbolic paraboloid structure used as a roof. The form follows a convex curve about one axis and a concave curve about the other....

The Journal of Hyperbolic Differential Equations was founded in 2004 and carries papers pertaining to nonlinear hyperbolic problems and related mathematical...

where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: cosh ⁡...

wants. Histrionic personality disorder is also linked with hyperbolic language and hyperbolic communication. People diagnosed with HPD may be dramatic....

hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively. For a right circular cylinder, there are several ways in...

Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet...

Picard theorem: maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained...

paper De solido hyperbolico acuto, written in 1643, a truncated acute hyperbolic solid, cut by a plane. Volume 1, part 1 of his Opera geometrica published...

A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is a special type of spatial network where (1) latent coordinates of nodes are...

Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jørgensen...

mathematics, the term hyperbolic plane may refer to: A two-dimensional plane in hyperbolic geometry A quadratic space known as the hyperbolic plane (quadratic...

algebraically hyperbolic if the special set is empty. Lang conjectured that a variety X is Mordellic if and only if X is algebraically hyperbolic and that...

this concept include: "non-numerical vague quantifier" and "indefinite hyperbolic numerals". Umpteen, umteen or umpty is an unspecified but large number...

goal of hyperbolic tapering is to achieve linear reductions of brain receptor occupancy. For example when tapering citalopram, a hyperbolic tapering...

torture and murder over the past 150 years, it might not be all that hyperbolic to say that in ISIS, what we see more than anything else is a more expansive...

doubt, methodological skepticism, universal doubt, systematic doubt, or hyperbolic doubt. Cartesian doubt is a systematic process of being skeptical about...

In hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by William Thurston (1986). Given a...

conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements. Let O denote the unknot. For any knot K...

number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those...

influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical Society...

In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron...

indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals...

instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as...

Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate...

When the real line is considered the boundary of the hyperbolic plane, PSL(2, R) expresses hyperbolic motions. Elements of PSL(2, R) act on the complex plane...

The law of hyperbolic growth of the human population is an empirical law discovered by Heinz von Foerster, which states that the human population of the...

and topology. He is known for his contributions to the understanding of hyperbolic 3-manifolds and the geometry of Teichmüller spaces. Since July 2018, Brock...

discovered in several spaces, as in hyperbolic space and spherical geometry. A triangle in hyperbolic space is called a hyperbolic triangle, and it can be obtained...

truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon...

{\displaystyle x} ⁠, and some algebra, we obtain a pair of damped, dispersive hyperbolic partial differential equations each involving only one unknown: ∂ 2 ∂...

shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit (which...

in hyperbolic (Lobachevskian) geometry", and because squaring the circle is seen as a "famous impossibility" despite being possible in hyperbolic geometry...

{\displaystyle S} to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for...

the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is...

diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field...

known as density of hyperbolicity, is one of the most important open problems in complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot...

compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants...

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...

between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include...

ellipsoids and hyperboloids are special limiting cases. Any real ellipsoid or hyperbolic and be expressed as a repositioned matrix of the form M = P T M ´ P =...

geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this...

Hyperbolic structure may refer to: Hyperboloid structure Hyperbolic set This disambiguation page lists mathematics articles associated with the same title...

fittingly dynamic homage." Among the critics was Time's Richard Corliss (hyperbolically) saying that Boseman "deserves a Pulitzer, a Nobel and instant election...

Quit 3 (2024) Dump Gawd: Hyperbolic Time Chamber Rap (2024) Dump Gawd: Hyperbolic Time Chamber Rap 2 (2024) Dump Gawd: Hyperbolic Time Chamber Rap 3 (2024)...

saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introduced hyperbolic quaternions to extend the...

is necessary. An easy example for an ε-quadratic form is the standard hyperbolic ε-quadratic form H ε ( R ) ∈ Q ε ( R ⊕ R ∗ ) {\displaystyle H_{\varepsilon...

approximating integrals Maximum and minimum Derivative test Hyperbolic equilibrium point Hyperbolic geometry Minimax theorem Max–min inequality Mountain pass...

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather...

In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron...

In geometry, the order-7 heptagrammic tiling is a tiling of the hyperbolic plane by overlapping heptagrams. This tiling is a regular star-tiling, and has...

union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure-eight...

maint: multiple names: authors list (link) Levin, Sam (August 11, 2025). "'Hyperbolic and false': Trump's portrayal of crime in Washington DC has little to...

light being infinite. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion,...

space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere...

field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space...

A catenary arch is a type of architectural arch that follows an inverted catenary curve. The catenary curve has been employed in buildings since ancient...

sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent...

C/2018 C2 (Lemmon) is a hyperbolic comet (previously classified as A/2018 C2, a hyperbolic asteroid). It was first observed on 5 February 2018 by the Mount...

was an English mathematician, known for Osborn’s rule that deals with hyperbolic trigonometric identities. Osborn was born in 1864 in Manchester, England...

curvature are defined. Euclidean space, the n {\displaystyle n} -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids...

two forms of non-Euclidean geometry. Although spherical geometry and hyperbolic geometry both lack polygons with four equal sides and right angles, they...

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...

foliation. Hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane)...

Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in...

priest, mathematician, and physicist. He is credited with introducing hyperbolic functions. Vincenzo Riccati was born in 1707 in Castelfranco Veneto, a...

a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits...

In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each...

Mikhail Gromov "Hyperbolic groups" that introduced the notion of a hyperbolic group (also known as word-hyperbolic or Gromov-hyperbolic or negatively curved...

Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one...

for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball...

Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. The tiles are congruent...

discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group...

In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3...

the second word of دالة زائدية "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to...

1956) is an American mathematician primarily working in the areas of hyperbolic 3-manifolds and knot theory. His book, The Knot Book, has been praised...

than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H3. Curved geometries are in the domain of non-Euclidean geometry...

finite index), or the hyperbolic plane. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A Fuchsian group...

prove that hyperbolic volume decreases under hyperbolic Dehn surgery. Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry, Universitext...

attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. The gravitational tidal forces acting...

Princeton University. His research focuses on low-dimensional topology and hyperbolic geometry. David Gabai received his B.S. in mathematics from MIT in 1976...

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with...

mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense of Gromov. Let (X, d) be a metric space and...

parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the three ridge lines passing through the umbilic and hyperbolic umbilics have just one...

compact and noncompact hyperbolic Dynkin graphs has been enumerated. All rank 3 hyperbolic graphs are compact. Compact hyperbolic Dynkin diagrams exist...

than the escape speed at its current distance. In contrast if it is on a hyperbolic trajectory its speed will always be higher than the escape speed at its...

(complete) hyperbolic structures on a finite volume hyperbolic n {\displaystyle n} -manifold (for n > 2 {\displaystyle n>2} ) is a point, for a hyperbolic surface...

In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with...

as limit groups over hyperbolic and certain relatively hyperbolic groups. Basic examples include free groups themselves, hyperbolic orientable surface groups...

{\displaystyle \operatorname {Ci} (x)=\gamma +\ln x-\operatorname {Cin} (x).} The hyperbolic sine integral is defined as Shi ⁡ ( x ) = ∫ 0 x sinh ⁡ t t d t . {\displaystyle...

\operatorname {sech} (n\theta )} where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are named after the French mathematician...

nonlinear wave propagation, Rational Extended Thermodynamics (RET), and hyperbolic formulations of general relativity. He graduated in Theoretical Physics...

spectral geometry, on the expansion and spectral properties of random hyperbolic surfaces, continues the work of Maryam Mirzakhani. She works in England...

calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One...

feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth...

In the geometry of hyperbolic 3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular...

become widespread in popular culture. The "facts" are normally absurd hyperbolic claims about Norris's toughness, attitude, virility, sophistication, and...

self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are...

sine law using elementary linear algebra and projection matrices. In hyperbolic geometry when the curvature is −1, the law of sines becomes sin ⁡ A sinh...

G. Petrovsky, is a region where the fundamental solution of a linear hyperbolic partial differential equation vanishes. They were studied by Petrovsky...

structure of cocompact discrete hyperbolic groups" was one of the forerunners in the development of the theory of word-hyperbolic groups, a notion that was...

In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite...

mathematician at the University of Fribourg, whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities. As a child...

Shukhov (1853–1939). Shukhov built his first example as a water tower (hyperbolic shell) for the 1896 All-Russian Exposition. Subsequently, more have been...

group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically...

In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space H 3 / Γ {\displaystyle \mathbb {H} ^{3}/\Gamma...

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame,...

In the geometry of hyperbolic 3-space, the cubic-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, cube, and cuboctahedron...

reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical...

This is a list of indefinite sums (also known as antidifferences) of various functions. An indefinite sum ∑ x f ( x ) {\textstyle \sum _{x}f(x)} is the...

In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic: one whose parametrization is not obtained by going repeatedly...

Two-dimensional spaces can also be curved, for example the sphere and hyperbolic plane, sufficiently small portions of which appear like the flat plane...

proved by Robert Geroch) gives a topological characterization of globally hyperbolic spacetimes. This theorem was improved by A.N. Bernal and M. Sánchez, who...

This is a list of parabolic and hyperbolic comets in the Solar System. Many of these comets may come from the Oort cloud, or perhaps even have interstellar...

field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It...

TOAs are multiple and known. When MLAT is used for navigation (as in hyperbolic navigation), the waves are transmitted by the stations and received by...

base space is zero. The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, given that the geometrization conjecture is now...

– inverse hyperbolic cosecant function. (Also written as arcsch.) arcosh – inverse hyperbolic cosine function. arcoth – inverse hyperbolic cotangent function...

The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals...

In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant...

Archimedean spiral Cornu spiral Cotes' spiral Fermat's spiral Galileo's spiral Hyperbolic spiral Lituus Logarithmic spiral Nielsen's spiral Golden spiral Syntractrix...

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces...

Euclidean plane 3-7 kisrhombille – hyperbolic plane 3-8 kisrhombille – hyperbolic plane 4-5 kisrhombille – hyperbolic plane Conway, John; Burgiel, Heidi;...

hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics, the coordinates of a hyperbolically accelerated...

In the geometry of hyperbolic 3-space, the cubic-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from cube, triangular tiling...

partner preferring women with "giant hips". Both descriptions serve to hyperbolically describe both the sexual preferences of the main characters, and their...

hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. One form of the hyperbolization...

In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite...

{\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.} A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument...

1934) is an American mathematician, specializing in complex analysis and hyperbolic geometry. He is a son of another prominent mathematician, Morris Marden...

Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an...

sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic...

v t e Knot theory (knots and links) Hyperbolic Figure-eight (41) Three-twist (52) Stevedore (61) 62 63 Endless (74) Carrick mat (818) Perko pair (10161)...

field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space...

Leonhard Euler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits by Johann Lambert in 1761–1777. Another milestone in orbit determination...

function. The main models of discounting include exponential, hyperbolic, and quasi hyperbolic. The higher the time preference, the higher the discount placed...

In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3...

Kahn (born October 26, 1969) is an American mathematician. He works on hyperbolic geometry, Riemann surfaces and complex dynamics. Kahn grew up in New York...

algorithm for hyperbolic interpolation that is used in GPUs as a perspective correct alternative to linear interpolation. Both the linear and hyperbolic variants...

being distorted by the Doppler effect. In a hyperbolic chirp, the frequency of the signal varies hyperbolically as a function of time: f ( t ) = f 0 f 1...

surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry...

There are also 5 compact hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of...

In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli...

example, when departing a planet, the spacecraft follows a hyperbolic escape trajectory with hyperbolic excess velocity v ∞ {\displaystyle \mathbf {v} _{\infty...

help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first...

hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which...

In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3...

11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex...

. The hyperbolic case is similar, with the area of a disk of intrinsic radius R in the (constant curvature − 1 {\displaystyle -1} ) hyperbolic plane given...

two geometries here real-hyperbolic 4-space H R 4 {\displaystyle \mathbf {H} _{\mathbb {R} }^{4}} and the complex hyperbolic plane H C 2 {\displaystyle...

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons...

put on the set H of hyperbolic 3-manifolds of finite volume. Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental...

In the geometry of hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists...

Kac–Moody algebras of indefinite type, most work has focused on those hyperbolic type, for which the matrix S is indefinite, but for each proper subset...

In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has...

include: The Archimedean spiral: r = a φ {\displaystyle r=a\varphi } The hyperbolic spiral: r = a / φ {\displaystyle r=a/\varphi } Fermat's spiral: r = a...

n-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous...

In hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} is the angle at the non-right angle vertex of a right hyperbolic triangle...

model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Blockmodeling Maximum entropy Soft...

songs depict grief and emotional tumult caused by ill-fated romance in a hyperbolic, confrontational, and self-aware manner, exploring themes of mourning...

the real projective plane. One may also conceive of a hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature. Abstractly, one...

utilization reached the least upper bound presented by Liu and Layland. The hyperbolic bound is a tighter sufficient condition for schedulability than the one...

definition of curvature, the theorem also applies in spherical geometry and hyperbolic geometry. In higher dimensions, an analogous quadratic equation applies...

place in the hyperbolic plane. HyperRogue is a turn-based game in which the player controls one character exploring a world based on hyperbolic geometry,...

most important implication of the hyperbolic equation is that by switching from a parabolic (dissipative) to a hyperbolic (includes a conservative term)...

dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an...

In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation (or honeycomb). With Schläfli symbol...

and probability theory. They are also the simplest examples of Gromov hyperbolic spaces. A metric space X {\displaystyle X} is a real tree if it is a geodesic...

In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a...

figure of 985 as the number of villages destroyed during the war seems hyperbolic, all Judaean villages, without exception, excavated thus far were razed...

orbits ( 0 ≤ e < 1 {\displaystyle 0\leq e<1} ). The hyperbolic Kepler equation is used for hyperbolic trajectories ( e > 1 {\displaystyle e>1} ). The radial...

elliptical orbits, undefined for parabolic trajectories, and negative for hyperbolic trajectories, which can hinder its usability when working with different...

convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time...

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with...

formal grammars Continuum hypothesis, in set theory Hyperbolic cosine, in mathematics, a hyperbolic function, ch(x) = cosh(x) Curry–Howard correspondence...

locally symmetric space. In the case of the Basel problem, it is the hyperbolic 3-manifold SL2(R)/SL2(Z). The zeta function also satisfies Riemann's functional...

relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the...

accelerating improvement, then levels off without continuing upward into a hyperbolic singularity. Alan Turing, often regarded as the father of modern computer...

angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal...

C/2017 U7 (PanSTARRS) is a hyperbolic comet (previously classified as A/2017 U7, a hyperbolic asteroid), first observed on 29 October 2017 by astronomers...

In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,3...

Euclidean plane ( E 2 {\displaystyle E^{2}} ), and their analogues on the hyperbolic plane ( H 2 {\displaystyle H^{2}} ). The following types of Euclidean...

potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics. Consider a polygon in the complex plane. The...

lowest known dimension for an exotic sphere is the seventh dimension. In hyperbolic space, 7 is the highest dimension for non-simplex hypercompact Vinberg...

{\displaystyle u\in H^{m+2}(U)} , which also holds if m is infinity. Parabolic and hyperbolic PDEs describe the time evolution of a quantity u governed by an elliptic...

systems, giving rise to Euclidean or hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry...

tessellation of the hyperbolic plane by right triangles with angles of 30°, 45°, and 90°; triangles with these angles are possible in hyperbolic geometry but...

Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio...

creations apply hyperbolic (curved) geometric shapes—distinguished from Euclidean (flat) geometry—to emulate natural structures. Extending hyperbolic crochet...

are geometries that do not satisfy the second form of the postulate. A hyperbolic geometry is a geometry that does not satisfy the original postulate. An...

known hyperbolic 3-manifolds, first described by Émile Picard in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space...

In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli...

is a Scottish-American mathematician working primarily with arithmetic hyperbolic 3-manifolds. He is the Edgar Odell Lovett Chair of mathematics at Rice...

article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces. For any natural number n, there are n-pointed star regular polygonal...

even number of hyperbolic sines. The Gudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does...

was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory. McMullen graduated as valedictorian in...

subgroup of the group of isometries of the hyperbolic plane. If we consider the upper half-plane model H of hyperbolic plane geometry, then the group of all...

{x}{2}}-{\frac {1}{5}}\tan ^{5}{\frac {x}{2}}+C.\end{aligned}}} Substitutions of hyperbolic functions can also be used to simplify integrals. For example, to integrate...

Pan-STARRS is the Panoramic Survey Telescope and Rapid Response System in Hawaii, United States Objects discovered by Pan-STARRS include: 311P/PanSTARRS...

In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal...

meant to convey an extremely short time such as an instant, sometimes in hyperbolic form, as in "perhaps faster than you would believe is possible," referring...

his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov...

construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. The Wythoff construction begins by choosing a generator point on...

transformations on the ideal boundary S 2 {\displaystyle \mathbb {S} ^{2}} of the hyperbolic 3-space H 3 {\displaystyle \mathbb {H} ^{3}} . The notion of a convergence...

of nearly spherical surfaces: umbilical point. Thom proposed that the hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical...

risk of suicide was managed and that Assange had a "self-dramatising or hyperbolic approach in describing symptoms". Patrick Eller, a former forensics examiner...

Retrieved 5 August 2008. The paper explores the assemblies of second order hyperbolic surfaces as they are used throughout the design composition of the Sagrada...

subtraction, multiplication, division) and advanced (trigonometric, hyperbolic, etc.) mathematical operations and functions. They have completely replaced...

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

calling it "almost certainly correct". However, reviewers criticised the hyperbolic claims made for the book. G. L. Huxley and Christina Haywood both criticised...

In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron...

In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and...

In the geometry of hyperbolic 3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling...

geometry, the cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}. John H. Conway...

identified as interstellar interlopers due to possessing significant hyperbolic excess velocity, indicating they did not originate in the solar system...

v t e Knot theory (knots and links) Hyperbolic Figure-eight (41) Three-twist (52) Stevedore (61) 62 63 Endless (74) Carrick mat (818) Perko pair (10161)...

}}}}}}}}.} The inverse hyperbolic functions are related to the inverse trigonometric functions similar to how the hyperbolic functions are related to...

In financial mathematics, tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk...

of distance in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be viewed as a disk as illustrated on the right. This...

Comet Gibbs may refer to any of the following comets discovered by American astronomer, Alex R. Gibbs, below: 313P/Gibbs 331P/Gibbs 229P/Gibbs 248P/Gibbs...

dt} \over {1+t^{2}}}.} One can play an entirely analogous game with the hyperbolic functions. A point on (the right branch of) a hyperbola is given by (cosh...

sungrazers, formed from the breakup of a single parent. Some comets with hyperbolic orbits may originate outside the Solar System, but determining their precise...

usually referred to as hyperbolic LCSs, as they provide a finite-time generalization of the classic concept of normally hyperbolic invariant manifolds in...

July 2026 through Etcetc Music. A follow-up to their previous album, Hyperbolic (2024), Ahhcade was supported by the release of the singles "Rollin'"...

weight system studied by Dror Bar-Natan. By numerical examinations on some hyperbolic knots, Rinat Kashaev discovered that substituting the n-th root of unity...

semi-check), There are also hyperbolic uniform honeycombs composed of only regular cells (Coxeter & Whitrow 1950), including: Hyperbolic uniform honeycombs, 3D...

In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists...

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2 q...

angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex (or more) and sphere with...

In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each...

Prize 2015 for Mathematical Sciences. He is best known for his work in hyperbolic geometry, geometric group theory, low-dimensional topology and complex...

Comet ATLAS may refer to any comets below discovered by the Asteroid Terrestrial-impact Last Alert System survey. The interstellar comet can be found at...

This hyperbolic geometry-related article is a stub. You can help Wikipedia by adding missing information....

constant negative scalar curvature. It is the Lorentzian analogue of hyperbolic space. Anti-de Sitter space and de Sitter space are named after Willem...

inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct...

of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations. These methods were developed from ENO...

As by definition of p one has this can be written For a hyperbolic orbit one uses the hyperbolic functions for the parameterisation for which one has and...

In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel. The ultraparallel theorem states...

called the Darboux problem) is a boundary value problem for a second-order hyperbolic partial differential equation (PDE) in two independent variables, with...

Maciejko, Rayan developed hyperbolic band theory, a proposed extension of Bloch theory from Euclidean crystals to hyperbolic lattices. The work was motivated...

In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston (1982) as the eleventh problem out of his twenty-four...

sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic...

LORAN (Long Range Navigation) was a hyperbolic radio navigation system developed in the United States during World War II. It was similar to the UK's Gee...

^{2}}{\partial \theta ^{2}}}f} A similar technique works in hyperbolic space. Here the hyperbolic space Hn−1 can be embedded into the n dimensional Minkowski...

precisely, a pleated surface is an isometry from a complete hyperbolic surface S to a hyperbolic 3-fold such that every point of S is in the interior of a...

equivalent to the parallel postulate, leading to Euclidean geometry. In hyperbolic geometry, arising from the negation of the parallel postulate, the summit...

looking at hyperbolic space. hyperbolic trigonometry the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry...

one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloid The degenerate quadric surfaces are the empty set, a single...

also inconsistent with the pictures that appear in: Alternating knot Hyperbolic knot Irrational winding of a torus Satellite knot Torus Knot on Wolfram...

under this diffeomorphism. Every hyperbolic component of the tricorn is simply connected. The boundary of every hyperbolic component of odd period of the...

Terrestrial-impact Last Alert System (ATLAS) station. The comet follows an unbound, hyperbolic trajectory past the Sun, and passed by Earth at 1.8 AU, posing no threat...

their DME signals can be used by civilian receivers.[citation needed] Hyperbolic navigation systems are a modified form of transponder systems which eliminate...

Precursors Galilean relativity Galilean transformation Aether theories Hyperbolic quaternions People Einstein Sommerfeld Michelson Morley FitzGerald Herglotz...

system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition...

of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane...

spectral geometry, and the theory of automorphic forms. In the case of hyperbolic surfaces, it translates information about the spectrum of the Laplace–Beltrami...

speculations of Lobachevski and Bolyai concerning hyperbolic geometry by providing models for the hyperbolic plane: for example, the Poincaré disc model where...

Escherich died in Vienna. Following Eugenio Beltrami's (1868) discussion of hyperbolic geometry, Escherich in 1874 published a paper named "The geometry on surfaces...

for his work in low-dimensional topology using computational tools and hyperbolic geometry, being one of the inspirations for William Thurston's later breakthroughs...

length 9 Braid no. 4 Bridge no. 2 Crosscap no. 2 Crossing no. 7 Genus 1 Hyperbolic volume 2.82812 Stick no. 9 Unknotting no. 1 Conway notation [52] A–B notation...

elliptic iff its mean affine curvature H > 0 {\displaystyle H>0} , and hyperbolic iff H < 0 {\displaystyle H<0} . All quadrics are affine spheres. Under...

a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his...

spherical and elliptic geometries, every pair of lines intersects, while in hyperbolic geometry there exist infinitely many distinct lines through a given point...

logarithmus hyperbolicus est = 1), … " ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … ) Remmert, Reinhold (1991)...

v t e Knot theory (knots and links) Hyperbolic Figure-eight (41) Three-twist (52) Stevedore (61) 62 63 Endless (74) Carrick mat (818) Perko pair (10161)...

known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence...

played early recordings from Beastie Boys. These segments feature the hyperbolic introductions of Noise The Show's host, Tim Sommer, an early supporter...

In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3...

alternating link is hyperbolic, i.e. the link complement has a hyperbolic geometry, unless the link is a torus link. Thus hyperbolic volume is an invariant...

punctured torus T {\displaystyle \mathbb {T} } with a complete, finite-volume hyperbolic structure is given by ∑ γ 1 1 + e ℓ ( γ ) = 1 2 {\displaystyle \sum _{\gamma...

(alternatively, "pigs might fly") is an adynaton—a figure of speech so hyperbolic that it describes an impossibility. The implication of such a phrase is...

many equal-length segments. The regular pseudogon is a partition of the hyperbolic line H1 (instead of the Euclidean line) into segments of length 2λ, as...

orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit Equations Dynamical friction Escape...

k-uniform tilings List of Euclidean uniform tilings Uniform tilings in hyperbolic plane Weisstein, Eric W. "Tessellation". mathworld.wolfram.com. Retrieved...

shared between the trigonometric functions and the hyperbolic functions, it is possible to use hyperbolic identities to construct a similar form of the substitution...

generalized circles in the complex plane. To construct models of the hyperbolic plane the unit disk and the upper half-plane are used to represent the...

of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-space). Roughly, it states that within a fixed radius, usually called...

{\displaystyle t=1} give the expected value of these basic trigonometric and hyperbolic functions over a Gaussian random variable X ∼ N ( μ , σ 2 ) {\displaystyle...

saying that the film was "approved to portray these facts in "colorful and hyperbolic" terms". On September 2, 2016, Jerry Heller died of a car accident, preceded...

mutations can occur from spherical tilings to Euclidean tilings to hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact and...

order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It covers the hyperbolic plane, which is a non-Euclidean surface with constant...

To uphold causality, Minkowski restricted spacetime to non-Euclidean hyperbolic geometry. Because signals and other causal influences cannot travel faster...

ideal points). The sequence extends when hyperbolic tilings are themselves used as facets of noncompact hyperbolic tessellations, as in the heptagonal tiling...

Lorentz boost and in mathematics it is a hyperbolic rotation. Each reference frame is associated with a hyperbolic angle, which is zero for the rest frame...

The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with...

about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearization—a natural simplification...

generation of humanists, both Catholic and Protestant, and the lurid hyperbolic attacks in his half-satire The Praise of Folly were later treated by Protestants...

generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's...

Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight...

reaches thermodynamic equilibrium. If the curvature of the universe is hyperbolic or flat, or if dark energy is a positive cosmological constant, the universe...

infinite skew polygons exist in the Petrie polygons of the affine and hyperbolic Coxeter groups. They are constructed a single operator as the composite...

In mathematics, cis is a function defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function...

speed divided by lightspeed c is the hyperbolic sine of rapidity η, just as the Lorentz factor γ is rapidity's hyperbolic cosine, and coordinate speed v over...

namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian...

=0} , as an example of this type. B2 − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions...

"AEIOU" with Empire of the Sun and announced their sixth studio album, Hyperbolic, which was released in March 2024. In May 2026, the duo announced their...

Institutions University of Wisconsin, Madison Thesis Periodic Solutions of Nonlinear Hyperbolic Differential Equations (1966) Doctoral advisor Jürgen Moser...

nature translated into his use of ruled geometrical forms such as the hyperbolic paraboloid, the hyperboloid, the helicoid and the cone, which reflect...

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons...

corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras...

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}. This tiling is topologically related...

hanging rope. Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the...

C/2023 V4 (Camarasa-Duszanowicz) is a hyperbolic comet that was discovered on November 5, 2023, by two astronomers Jordi Camarasa from Sabadell in Spain...

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

of angles. Lorentz boosts act by hyperbolic rotation of a spacetime plane, and such "rotations" preserve hyperbolic angle, the measure of rapidity used...

reflector as the primary while the secondary mirror is hyperbolic. Modern variants may have a hyperbolic primary for increased performance (for example, the...

non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the...

velocity. Radial hyperbolic orbit: An open hyperbolic orbit where the object is moving at greater than the escape velocity. This is a hyperbolic orbit with...

This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here...

The band model is a conformal model of the hyperbolic plane. The band model employs a portion of the Euclidean plane between two parallel lines. Distance...

familiar transcendental functions are the exponential, trigonometric, and hyperbolic functions, and their inverses, such as the logarithm and inverse trigonometric...

In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: ◻ {\displaystyle \Box } ), also called the d'Alembertian...

integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other...

unpublished work, which today we call hyperbolic geometry. Several authors still consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. In...

and several stereotypes and tropes of unscripted reality shows, with hyperbolic emphasis on exaggerated dramatic elements, and canned infighting among...

manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead...

On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations. The Euler equations can be formulated...

have concluded that biblical descriptions of violence are few and mostly hyperbolic, based on comparisons to the literary styles of ancient Israel's neighbouring...

Sydler's result to spherical or hyperbolic geometry remains open: it is not known whether two spherical or hyperbolic polyhedra with the same volume and...

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}. The symmetry group of the tiling...

metric spaces. After Eliyahu Rips applied the same complex to the study of hyperbolic groups, its use was popularized by Mikhail Gromov (1987), who called it...

will keep slowing it down but not towards zero. It will approach the hyperbolic excess speed v ∞ = v 2 − v e 2 . {\displaystyle v_{\infty }={\sqrt...

Comet Leonard may refer to any of the comets discovered by Gregory J. Leonard below: 396P/Leonard 449P/Leonard 486P/Leonard P/2018 VN2 (Leonard) P/2021...

trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions. The fundamental problem of symbolic integration...

angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right...

An iteration of nine squeeze mappings applied to a hyperbolic sector...

relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory. This article provides a...

an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise...

its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces...

however, two other self-consistent non-Euclidean geometries are known, hyperbolic and elliptic geometry. An implication of Albert Einstein's theory of general...

Salsa, and Putine open a portal to a pocket dimension similar to the Hyperbolic Time Chamber, which will give Goku and Vegeta some time to train and master...

truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}. The dual of this tiling represents...

tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}. This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining...

tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}. This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting...

some scholars to believe that the fearful reports from the other ten are hyperbolic and should not be taken literally. Joshua finally expelled the Anakim...

specialized variant of the Cassegrain telescope that has a hyperbolic primary mirror and a hyperbolic secondary mirror designed to eliminate off-axis optical...

Jørgensen is a Danish mathematician at Columbia University working on hyperbolic geometry and complex analysis, who proved Jørgensen's inequality. He wrote...

the hyperboloid model of hyperbolic space already known in the 19th century, because isometries (or motions) in hyperbolic space can be related to Lorentz...

including Jewish New Yorkers." The Nexus Project called the ADL "divisive, hyperbolic and aggressive" for planning to monitor Mamdani's administration for antisemitism...

dodecadodecahedron. It is topologically equivalent to a quotient space of the hyperbolic order-4 pentagonal tiling, by distorting the pentagrams back into regular...

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called...

Comet NEAT may refer to any comets below discovered by the Near-Earth Asteroid Tracking survey: 163P/NEAT 169P/NEAT 180P/NEAT 189P/NEAT 207P/NEAT 212P/NEAT...

portrayed the prosecutor, Attorney General Gideon Hausner, as employing hyperbolic rhetoric in the pursuit of Prime Minister Ben-Gurion's political agenda...

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}. A half symmetry [1+,4...

In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has...

astronomical objects List of Solar System objects by greatest aphelion List of hyperbolic comets List of trans-Neptunian objects "MPEC 2021-C187 : 2018 AG37". Minor...

Noah serving as executive producers. The show intends to "satirize the hyperbolic, conspiracy-laden noise machine that is the alternative-media landscape...

In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism...

sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic...

model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Blockmodeling Maximum entropy Soft...

{\displaystyle f_{n}^{\prime }} is defined, then one says that a periodic point is hyperbolic if | f n ′ | ≠ 1 , {\displaystyle |f_{n}^{\prime }|\neq 1,} that it is...

unitary group SU(1,1), which is used to describe hyperbolic motions of the Poincaré disk model in hyperbolic geometry. Split-quaternions may be generated...

(born 24 March 1951) is an English mathematician known for her work in hyperbolic geometry, Kleinian groups and dynamical systems. Series was born on March...

called hyperbolic. If ℓX(G) ≠ 0 then there exists a unique minimal G-invariant subtree XG of X. Moreover, XG is equal to the union of axes of hyperbolic elements...

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is paracompact because...

model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Blockmodeling Maximum entropy Soft...

model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Blockmodeling Maximum entropy Soft...

also known: the problem is to identify the positions for all sensors. Hyperbolic navigation is one pre-GPS technology that uses distance geometry for locating...

Comet Lemmon may refer to any comets below discovered by the Mount Lemmon Survey: 349P/Lemmon P/2023 JN16 (Lemmon) 363P/Lemmon 397P/Lemmon 405P/Lemmon...

model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG) Hyperbolic (HGN) Hierarchical Stochastic block Blockmodeling Maximum entropy Soft...

lawmakers. Publicly, he used the media and speaking engagements to introduce hyperbolic messages about the evils of drug use. In the 1930s, he referred to a collection...

{s-a}{2}}\tan {\frac {s-b}{2}}\tan {\frac {s-c}{2}}} For a triangle in hyperbolic geometry the analogous formula is tan 2 ⁡ S 4 = tanh ⁡ s 2 tanh ⁡ s −...

advent of shell construction and the better mathematical understanding of hyperbolic paraboloids allowed very thin, strong vaults to be constructed with previously...

parabolic trajectories rv2 is constant and equal to 2μ. For elliptic and hyperbolic orbits magnitude of μ = 2 times the magnitude of a times the magnitude...

line parallel to the base. An analogous theorem can also be proven for hyperbolic triangles, for which the apex lies on a hypercycle. Given a fixed base...

mirror is reflected such that it passes through the other. The focus of a hyperbolic mirror is either of two points which have the property that light from...

In 2-dimensional hyperbolic geometry, the infinite-order hexagonal tiling is a regular tiling. It has Schläfli symbol of {6,∞}. All vertices are ideal...

some discretised space-time computational domain. A non-linear system of hyperbolic partial differential equations representing a set of conservation laws...

true without any doubt. To achieve this, he employed a method called hyperbolic or metaphysical doubt, also sometimes referred to as methodological skepticism...

In the geometry of hyperbolic 4-space, the cubic honeycomb honeycomb is one of two paracompact regular space-filling tessellations (or honeycombs). It...

it contains a free subgroup with two generators of index 12. A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or...

In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron...

trigonometric functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of exponential functions...

Shown from top to bottom are a closed universe with positive curvature, a hyperbolic universe with negative curvature and a flat universe with zero curvature...

and 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial...

In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron...

Zeitschrift. 65: 133–170. doi:10.1007/bf01473875. Purcell, Jessica (2020). Hyperbolic knot theory. American Mathematical Society. ISBN 978-1-4704-5499-9. Table...

In finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) refers to a type of risk aversion that is particularly convenient...

discuss] by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry. The nth Lagrange number can be calculated from the nth Markov...

eccentricity vector. It is strictly defined for elliptic, parabolic, and hyperbolic orbits. For such orbits the apse line is found: for elliptical orbits –...

and θ 8 {\displaystyle \theta ^{8}} gives some deep identities for the hyperbolic secant. In 1918, Hardy and Ramanujan studied the partition function P(n)...

unit disk forms the set of points for the Poincaré disk model of the hyperbolic plane. Circular arcs perpendicular to the unit circle form the "lines"...

hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk...

x_{i}x_{j}}{1+\delta x_{i}x_{j}}}.} Assuming that a network has an underlying hyperbolic geometry, one can use the framework of spatial networks to generate scale-free...

Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such...

\!} is the length of the semi-major axis. The velocity equation for a hyperbolic trajectory has either ( + 1 a ) {\displaystyle (+{1 \over {a}})} , or...

rotations. All these transformations are conformal since hyperbolic rotations preserve hyperbolic angle, (called rapidity) and the other rotations preserve...

physical spacetime will satisfy the strongest causality condition: global hyperbolicity. For such spacetimes the equations in general relativity can be posed...

argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle. A mathematical function has one or more arguments...

arising from resource limitation. The hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth...

There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations...

cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric...

Bernard Alper Epstein (born 1937) is a mathematician known for his work in hyperbolic geometry, 3-manifolds, and group theory, amongst other fields. He co-founded...

z-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficient r is near 1...

lattice Concept map Conceptual graph Decision tree Dendrogram Graph drawing Hyperbolic tree Hypertext Issue map Issue tree Layered graph drawing Mind map Object–role...

Chameleon Run is a 2016 action game developed by Hyperbolic Magnetism and published by Noodlecake Studios. It was released for iOS, Android, tvOS, Windows...

is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite difference method...

the form x − α , {\displaystyle x^{-\alpha },} of which the simplest is hyperbolic growth, where the exponent is (negative) 1: x − 1 . {\displaystyle x^{-1}...

configuration are hyperbolically rotated with respect to each other. The natural functions for expressing these relationships are the hyperbolic analogs of the...

solids, Klein bottles and Boy's surface. The Lorenz manifold and the hyperbolic plane have been crafted using crochet. Knitted and crocheted tori have...

research dealt with low-dimensional geometry and topology, Kleinian groups, hyperbolic geometry, geometric group theory, geometric representation theory in Lie...

Precursors Galilean relativity Galilean transformation Aether theories Hyperbolic quaternions People Einstein Sommerfeld Michelson Morley FitzGerald Herglotz...

special case of a Lambert quadrilateral. The fourth angle is acute in hyperbolic geometry and obtuse in spherical geometry. Every kite is an orthodiagonal...

rapidity, and the ordinary sense of simultaneity becomes dependent on hyperbolic orthogonality of spatial directions to the worldline associated to the...

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular...

non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite...

Horowitz suggests hyperbolic tapering, where the dose reductions become progressively smaller over time. For citalopram, a hyperbolic tapering regimen...

of the ellipse Elliptic catenary Line Hyperbola Focus of the hyperbola Hyperbolic catenary Line Rectangular hyperbola Center of the hyperbola Rectangular...

form of a logistic curve. The logistic function is an offset and scaled hyperbolic tangent function: f ( x ) = 1 2 + 1 2 tanh ⁡ ( x 2 ) , {\displaystyle...

Internet meme widespread in popular culture. These "facts" are absurd, hyperbolic claims about Norris's skills, toughness, attitude, sophistication, and...

Shibata. She is known for her research on mixed boundary conditions for hyperbolic partial differential equations, for which she won the 1974 Iyanaga Prize...

special relativity Fundamental concepts Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper...

of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently Möbius transformations of the Riemann sphere...

hallucinatory realm fully detached from reality, being typically characterised by hyperbolic geometry, and described as defying visual or verbal description. Users...

If the unknown x occurs only in linear expressions inside arguments of hyperbolic functions, unfolding them by their defining exponential expressions and...