All articles starting with H

Hyperbolic 3-manifold

A hyperbolic 3-manifold is a three-dimensional manifold with a complete Riemannian metric of constant negative curvature, the central object of William Thurston's 1980s geometrization program in topology.

Hyperbolic discounting

Hyperbolic discounting is a behavioral economics model of intertemporal choice in which the discount rate declines with delay, so people discount the near future steeply and the distant future gently, producing present bias and time-inconsistent preferences.

Hyperbolic functions

The hyperbolic functions, also called hyperbolic trigonometric functions, are elementary functions built from exponentials that play the role for the hyperbola that sine and cosine play for the circle, describing hanging cables and rapidity in relativity.

Hyperbolic geometry

Hyperbolic geometry, also called Lobachevsky geometry, is a non-Euclidean geometry with constant negative curvature in which triangles have angle sums below 180°, discovered independently by Lobachevsky and Bolyai in the 1820s.

Hyperbolic group

A hyperbolic group, also called a word-hyperbolic group, is a finitely generated group whose Cayley graph with the word metric is hyperbolic, meaning geodesic triangles are uniformly thin; introduced by Mikhail Gromov in 1987.

Hyperbolic lattices

Hyperbolic lattices are regular tessellations of the two-dimensional hyperbolic plane, a non-Euclidean space of uniform negative curvature, used as synthetic platforms for band theory, topology, and disorder physics.

Hyperbolic manifold

A hyperbolic manifold is a Riemannian manifold locally isometric to hyperbolic space of constant curvature -1, playing a central role in 3-dimensional topology and geometrization.

Hyperbolic metamaterial

A hyperbolic metamaterial is an artificial uniaxial optical material with opposite-sign permittivity components, giving an open hyperboloid dispersion that supports high-k waves for sub-diffraction imaging.

Hyperbolic metasurfaces

Hyperbolic metasurfaces are two-dimensional, subwavelength-structured surfaces whose surface plasmon polaritons have hyperbolic dispersion, enabling strong light confinement, negative refraction, and applications in imaging, emission control, and on-chip optics.

Hyperbolic motion (relativity)

Hyperbolic motion in relativity is the rectilinear motion of a body whose constant proper acceleration gives it a hyperbolic worldline, first studied by Max Born and underlying the Rindler horizon.

Hyperbolic paraboloid

A hyperbolic paraboloid is a saddle-shaped quadric surface, one of only two doubly ruled surfaces in three-dimensional space, with two families of straight lines lying entirely on it.

Hyperbolic phonon polaritons

Hyperbolic phonon polaritons are quasiparticles formed when infrared photons couple to lattice vibrations in polar crystals, confining light to scales far below its free-space wavelength with low loss.

Hyperbolic phonon polaritons in hexagonal boron nitride

Hyperbolic phonon polaritons in hexagonal boron nitride (HPhPs) are hybrid light-matter quasiparticles formed when infrared photons couple to optical phonons in the layered crystal hexagonal boron nitride.

Hyperbolic plane

The hyperbolic plane is the unique complete, simply connected surface of constant Gauss curvature −1, studied through models like the Poincaré disk because it cannot sit in ordinary 3-space.

Hyperbolic plasmons

Hyperbolic plasmons are surface plasmon polaritons whose isofrequency contour is a hyperbola rather than an ellipse, giving highly directional propagation and an extremely large density of states.

Hyperbolic polariton

A hyperbolic polariton is a hybrid light–matter quasiparticle that propagates in anisotropic media with permittivity components of opposite sign, giving hyperbolic dispersion, high confinement, and directional energy flow.

Hyperbolic polariton ray optics

Hyperbolic polariton ray optics treats light-matter waves called hyperbolic polaritons as rays, like geometrical optics, in anisotropic crystals such as hexagonal boron nitride and molybdenum trioxide.

Hyperbolic shear polaritons

Hyperbolic shear polaritons are hybrid light-matter quasiparticles arising in low-symmetry monoclinic and triclinic crystals, first demonstrated in beta-phase gallium oxide and reported in Nature in February 2022.

Hyperbolic surface

A hyperbolic surface is a two-dimensional surface with constant Gaussian curvature −1, locally like the hyperbolic plane, existing on closed orientable surfaces of genus at least 2.

Hyperbolic trajectory

A hyperbolic trajectory, also called a hyperbolic orbit, is the open path of a body moving under gravity with more than escape energy, corresponding to eccentricity greater than 1.

Hyperborea

Hyperborea (Ὑπερβόρεα), land of the Hyperboreans, a mythical blessed realm in Greek mythology beyond the north wind Boreas, tied to Apollo's cult at Delos and Delphi, first attested in Hesiod's Catalogue.

Hyperborea Lingula

Hyperborea Lingula: tongue-shaped plateau (lingula) at the mouth of Chasma Boreale on Mars, 124.80 km across, rising 200-350 m above Vastitas Borealis; IAU-named 2006, formed of the Rupes Tenuis unit.

Hyperboreae Undae

Hyperboreae Undae is a 463.65-km-wide dune field (unda) in Chasma Boreale near Mars's north pole, IAU-approved in 1988 from a classical albedo feature name, with coexisting barchan and linear dunes equilibrated to modern winds.

Hyperborei Cavi

Hyperborei Cavi: IAU-approved (1988) group of elongate steep-sided depressions, a cavus feature, cutting eastern Hyperborea Lingula near Mars' north pole; 92.81 km across, centered at 79.91°N, 310.30°E, a classical albedo feature name.

Hyperboreus Labyrinthus

Hyperboreus Labyrinthus is a Martian labyrinthus, a complex of intersecting valleys and ridges about 112 km across near 80°N, just south of the north polar cap; IAU-approved 1988 as a classical albedo feature name.

Hypercalcaemia

Hypercalcaemia (hypercalcemia): elevated blood calcium, defined as >10.5 mg/dL or adjusted calcium ≥2.6 mmol/L. Mostly caused by primary hyperparathyroidism and malignancy; treated with hydration, bisphosphonates, or denosumab.

Hypercalciuria

Hypercalciuria (ICD-11 MF98.0), excessive urinary calcium excretion from intestinal overabsorption, bone resorption, or renal tubular defects; the leading risk factor for calcium kidney stones, treated with thiazides, low-sodium diet, and normal calcium intake.

Hypercapnia

Hypercapnia (hypercarbia): elevated arterial carbon dioxide (PaCO2 >45 mm Hg) from alveolar hypoventilation; causes include COPD, obesity hypoventilation, neuromuscular disease; treated with NIV, controlled oxygen.

HyperCard

HyperCard was a hypermedia authoring and database program for the Macintosh, created by Bill Atkinson, released by Apple in 1987, and organized into cards and stacks.