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Type of non-Euclidean geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
Two geometries based on axioms closely related to those specifying Euclidean geometry
with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When isotropic
Non-Euclidean_geometry
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
Triangle in hyperbolic geometry
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
Hyperbolic_triangle
Concept in mathematics
Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the
Hyperbolic_metric_space
Geometry without the parallel postulate
of absolute geometry hold in hyperbolic geometry, which is a non-Euclidean geometry, as well as in Euclidean geometry. Absolute geometry is inconsistent
Absolute_geometry
Non-Euclidean geometry
the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves
Hyperbolic_space
24 mathematical problems stated in 1982
geometry posed by American mathematician William Thurston in his influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
Thurston's_24_questions
Hyperbolic analogues of trigonometric functions
sinh(t). Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations
Hyperbolic_functions
Region of the Cartesian plane bounded by a hyperbola and two radii
y={\sqrt {1+x^{2}}}} . The area of such hyperbolic sectors has been used to define hyperbolic distance in a geometry textbook. Squeeze mapping Augustus De
Hyperbolic_sector
Mathematical concept
satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987)
Hyperbolic_group
2009 book by Daina Taimina
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
Crocheting Adventures with Hyperbolic Planes
Crocheting_Adventures_with_Hyperbolic_Planes
Topics referred to by the same term
of smooth curve lying in a plane in mathematics Hyperbolic geometry, a non-Euclidean geometry Hyperbolic functions, analogues of ordinary trigonometric
Hyperbolic
Study of geometries as axiomatic systems
geometry hold in hyperbolic geometry as well as in Euclidean geometry. Absolute geometry is inconsistent with elliptic geometry: in elliptic geometry
Foundations_of_geometry
Isometric automorphisms of a hyperbolic space
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous
Hyperbolic_motion
hyperbolic space. hyperbolic trigonometry the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry. Other forms
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Relation used in geometry
affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines
Parallel_(geometry)
2D surface which extends indefinitely
real projective plane. One may also conceive of a hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature. Abstractly, one may
Plane_(mathematics)
Model of hyperbolic geometry
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside
Poincaré_disk_model
Type of mathematical link
of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component. As a consequence of the work
Hyperbolic_link
Branch of mathematics
spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky
Differential_geometry
Category of coordinate systems
plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
Dutch graphic artist (1898–1972)
reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical
M._C._Escher
Branch of mathematics
with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When isotropic
Geometry
Triangle area in terms of side lengths
{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 − a 2
Heron's_formula
Latvian mathematician
mathematics at Cornell University, known for developing a way of modeling hyperbolic geometry with crocheted objects. Taimiņa received all of her formal education
Daina_Taimiņa
Mathematical tree in the hyperbolic plane
A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying
Hyperbolic_tree
Trigonometric result for hyperbolic triangles
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
Hyperbolic_law_of_cosines
Overview of and topical guide to geometry
Elliptic geometry Enumerative geometry Epipolar geometry Euclidean geometry Finite geometry Fractal geometry Geometry of numbers Hyperbolic geometry Incidence
Outline_of_geometry
Three-holed sphere
surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are
Pair_of_pants_(mathematics)
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed
Arithmetic hyperbolic 3-manifold
Arithmetic_hyperbolic_3-manifold
Austrian mathematician
Eugenio Beltrami's (1868) discussion of hyperbolic geometry, Escherich in 1874 published a paper named "The geometry on surfaces of constant negative curvature"
Gustav_von_Escherich
Common point(s) shared by two lines in Euclidean geometry
explanation needed] In spherical and elliptic geometries, every pair of lines intersects, while in hyperbolic geometry there exist infinitely many distinct lines
Line–line_intersection
Three dimensional analogue of uniformization conjecture
one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological
Geometrization_conjecture
Topics referred to by the same term
In mathematics, hyperbolic trigonometry can mean: The study of hyperbolic triangles in hyperbolic geometry (traditional trigonometry is the study of triangles
Hyperbolic_trigonometry
Argument of the hyperbolic functions
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
Hyperbolic_angle
Manifold of dimension 3 equipped with a hyperbolic metric
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
Hyperbolic_3-manifold
Orientation-preserving mapping class group of the torus
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 orientation-preserving
Modular_group
Theory in number theory
mono-anabelian geometry in their absolute form. Shinichi Mochizuki also introduced combinatorial anabelian geometry which deals with issues of hyperbolic curves
Anabelian_geometry
Study of mathematics itself
discovery of hyperbolic geometry had important philosophical consequences for metamathematics. Before its discovery there was just one geometry and mathematics;
Metamathematics
Type of curve in hyperbolic geometry
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight
Hypercycle_(geometry)
Group realized geometrically by reflections across the sides of a triangle
type of the geometry by the Gauss–Bonnet theorem: it is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly
Triangle_group
Non-Euclidean geometry
stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from
Elliptic_geometry
Swiss mathematician
mathematician at the University of Fribourg, whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities. As a child
Ruth_Kellerhals
Covering by shapes without overlaps or gaps
often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed
Tessellation
Geometry without using coordinates
discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective
Synthetic_geometry
Mathematical functions
common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse
Inverse_hyperbolic_functions
these shapes Hilbert's third problem for non-Euclidean geometries: in spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
German mathematician (1905–1988)
Fenchel's duality theorem Geometry Convex geometry Brunn–Minkowski theorem Differential geometry Fenchel's theorem Hyperbolic geometry Jakob Nielsen Fenchel–Nielsen
Werner_Fenchel
Complex numbers with non-negative imaginary part
in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric
Upper_half-plane
Geometric axiom
Non-Euclidean geometries are geometries that do not satisfy the second form of the postulate. A hyperbolic geometry is a geometry that does not satisfy the
Parallel_postulate
1959 woodcut by M. C. Escher
of the hyperbolic plane by right triangles with angles of 30°, 45°, and 90°; triangles with these angles are possible in hyperbolic geometry but not
Circle_Limit_III
Type of geometry
speculations of Lobachevski and Bolyai concerning hyperbolic geometry by providing models for the hyperbolic plane: for example, the Poincaré disc model where
Projective_geometry
Mathematical space
diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory
3-manifold
Model of hyperbolic geometry
geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry
Beltrami–Klein_model
Upper-half plane model of hyperbolic non-Euclidean geometry
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically
Poincaré_half-plane_model
American mathematician (born 1958)
awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory. McMullen graduated as valedictorian in 1980
Curtis_T._McMullen
Study of angle-preserving transformations
transformation geometry soon appreciates the significance of Felix Klein's Erlangen program, an outgrowth of certain models of hyperbolic geometry. The combination
Inversive_geometry
American columnist, author and lecturer (born 1946)
being possible in hyperbolic geometry, then "if we reject a hyperbolic method of squaring the circle, we should also reject a hyperbolic proof of Fermat's
Marilyn_vos_Savant
Mathematics of smooth surfaces
analytic models for what Klein dubbed hyperbolic geometry. The four models of 2-dimensional hyperbolic geometry that emerged were: the Beltrami-Klein
Differential geometry of surfaces
Differential_geometry_of_surfaces
Group of real 2×2 matrices with unit determinant
Anosov flow Kisil, Vladimir V. (2012). Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R). London: Imperial College
SL2(R)
Bloch and Andrei Suslin. It is closely related to polylogarithm, hyperbolic geometry and algebraic K-theory. The dilogarithm function is the function
Bloch_group
Equation for radii of tangent circles
definition of curvature, the theorem also applies in spherical geometry and hyperbolic geometry. In higher dimensions, an analogous quadratic equation applies
Descartes's_theorem
Indian mathematician and monk of the Ramakrishna Order (born 1968)
is best known for his work in hyperbolic geometry, geometric group theory, low-dimensional topology and complex geometry. Mahan Mitra studied at St. Xavier's
Mahan_Mj
CLASS LECTURE: HYPERBOLIC GEOMETRY" (PDF). Retrieved August 12, 2018. Models of hyperbolic geometry Conformal Models of the Hyperbolic Geometry v t e
Band_model
Curve whose normals converge asymptotically
In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant
Horocycle
Concept in geometry
vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points. Unlike Euclidean and elliptic geometries, each line has two
Point_at_infinity
mostly describe basic background material on hyperbolic geometry. Chapter 4 covers Dehn surgery on hyperbolic manifolds Chapter 5 covers results related
The geometry and topology of three-manifolds
The_geometry_and_topology_of_three-manifolds
Shape with three sides
In non-Euclidean geometries, three "straight" segments also determine a "triangle", for instance, a spherical triangle or hyperbolic triangle. A geodesic
Triangle
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
Complex_hyperbolic_space
Russian mathematician (1792–1856)
and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet
Nikolai_Lobachevsky
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
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 {\displaystyle
Volume_conjecture
Shape with four equal sides and angles
balls for taxicab geometry and Chebyshev distance, two forms of non-Euclidean geometry. Although spherical geometry and hyperbolic geometry both lack polygons
Square
Möbius transformation generalized to rings other than the complex numbers
Uncertainty" [2] Kisil, Vladimir V. (2012). Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R). London: Imperial College
Linear fractional transformation
Linear_fractional_transformation
Geometric surface
was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry. By "the pseudosphere", people usually mean the tractroid. The tractroid
Pseudosphere
Quadrilateral with only 3 right angles
depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle
Lambert_quadrilateral
Model of n-dimensional hyperbolic geometry
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which
Hyperboloid_model
Symmetric subdivision in hyperbolic geometry
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Invariant in projective geometry
quadruple of points on the Riemann sphere. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio
Cross-ratio
Topological complexity in mathematics
prove that hyperbolic volume decreases under hyperbolic Dehn surgery. Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry, Universitext
Simplicial_volume
A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is a special type of spatial network where (1) latent coordinates of nodes are
Hyperbolic_geometric_graph
Hypersurface in hyperbolic space
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
Horosphere
Euclidean geometry without distance and angles
In the Minkowski geometry, lines that are hyperbolic-orthogonal remain in that relation when the plane is subjected to hyperbolic rotation. An axiomatic
Affine_geometry
Point at infinity in hyperbolic geometry
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
Ideal_point
Convex polyhedron with regular faces
Khovanskii, Geometry of generalized virtual polyhedra, J. Math. Sciences 269 (2023), 256–269. Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In
Johnson_solid
Form of geometry without distances
measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective
Ordered_geometry
Three linked but pairwise separated rings
produced in 1991 by the Geometry Center. Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner
Borromean_rings
Area in mathematics devoted to the study of finitely generated groups
with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. There are also substantial
Geometric_group_theory
Relation of space and time in relativity theory
In geometry, given a pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described
Hyperbolic_orthogonality
Set of points equidistant from a center
sphere. Spherical geometry is a form of elliptic geometry, which together with hyperbolic geometry makes up non-Euclidean geometry. The sphere is a smooth
Sphere
Fundamental result in geometry
foliation. Hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane)
Sum_of_angles_of_a_triangle
Equation used in relativistic physics
accounted in terms of the hyperbolic tangent function tanh which takes hyperbolic angle (rapidity) as an argument. In fact, the hyperbolic tangent of rapidity
Velocity-addition_formula
English mathematician (born 1951)
24 March 1951) is an English mathematician known for her work in hyperbolic geometry, Kleinian groups and dynamical systems. Series was born on March
Caroline_Series
German mathematician (1794–1874)
angles (which is now called hyperbolic geometry). While Schweikart never published his work (which he called "astral geometry"), he sent a short summary
Franz_Taurinus
Relation between sides of a right triangle
> c2. Here two cases of non-Euclidean geometry are considered—spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case
Pythagorean_theorem
Type of geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous
Klein_geometry
Independent video game
the hyperbolic plane. HyperRogue is a turn-based game in which the player controls one character exploring a world based on hyperbolic geometry, with
HyperRogue
American mathematician
the focus of Cannon's work shifted to the study of 3-manifolds, hyperbolic geometry and Kleinian groups and he is considered one of the key figures in
James_W._Cannon
Type of unbounded quadratic surface-shaped building or work
as well as the full set of supporting calculations analyzing the hyperbolic geometry and sizing the network of members, was completed by February 1919
Hyperboloid_structure
Geometric mean and hyperbolic angle as coordinates in quadrant I
carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence Q ↔ H P {\displaystyle Q\leftrightarrow
Hyperbolic_coordinates
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