# Horosphere > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Horosphere > Markdown URL: https://mediated.wiki/source/Horosphere.md > Source: https://en.wikipedia.org/wiki/Horosphere > Source revision: 1243622280 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Hypersurface in hyperbolic space}} [[File:633 honeycomb one cell horosphere.png|220px|right|thumb|A horosphere within the [PoincarĂ© disk model](/source/Poincar%C3%A9_disk_model) tangent to the edges of a [hexagonal tiling](/source/hexagonal_tiling) cell of a [hexagonal tiling honeycomb](/source/hexagonal_tiling_honeycomb)]] [[File:Apollonian spheres.jpg|thumb|[Apollonian sphere packing](/source/Apollonian_sphere_packing) can be seen as showing horospheres that are tangent to an outer sphere of a [PoincarĂ© disk model](/source/Poincar%C3%A9_disk_model)]] In [hyperbolic geometry](/source/hyperbolic_geometry), a '''horosphere''' (or '''parasphere''') is a specific [hypersurface](/source/hypersurface) in [hyperbolic ''n''-space](/source/hyperbolic_space). It is the boundary of a '''horoball''', the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For ''n'' = 2 a horosphere is called a [horocycle](/source/horocycle). A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane, but in hyperbolic geometry it is a horosphere (a curved surface). ==History== The concept has its roots in a notion expressed by [F. L. Wachter](/source/Friedrich_Ludwig_Wachter) in 1816 in a letter to his teacher [Gauss](/source/Carl_Friedrich_Gauss). Noting that in Euclidean geometry the limit of a sphere as its radius tends to infinity is a plane, Wachter affirmed that even if the [fifth postulate](/source/Euclid's_fifth_postulate) were false, there would nevertheless be a geometry on the surface identical with that of the ordinary plane.Roberto Bonola (1906), ''Non-Euclidean Geometry'', translated by [H.S. Carslaw](/source/Horatio_Scott_Carslaw), Dover, 1955; p. 63 The terms ''horosphere'' and ''horocycle'' are due to [Lobachevsky](/source/Nikolai_Lobachevsky), who established various results showing that the geometry of horocycles and the horosphere in hyperbolic space were equivalent to those of lines and the plane in Euclidean space.Roberto Bonola (1906), ''Non-Euclidean Geometry'', translated by H.S. Carslaw, Dover, 1955; p. 88 The term "horoball" is due to [William Thurston](/source/William_Thurston), who used it in his work on [hyperbolic 3-manifold](/source/hyperbolic_3-manifold)s. The terms horosphere and horoball are often used in 3-dimensional hyperbolic geometry. ==Models== In the [conformal ball model](/source/Poincar%C3%A9_disc), a horosphere is represented by a sphere tangent to the horizon sphere. In the [upper half-space model](/source/Poincar%C3%A9_half-plane_model), a horosphere can appear either as a sphere tangent to the horizon plane, or as a plane parallel to the horizon plane. In the [hyperboloid model](/source/hyperboloid_model), a horosphere is represented by a plane whose normal lies in the asymptotic cone. ==Curvature== A horosphere has a critical amount of (isotropic) curvature: if the curvature were any greater, the surface would close, yielding a sphere, and if the curvature were any less, the surface would be an (''N'' âˆ’ 1)-dimensional [hypercycle](/source/hypercycle_(geometry)). ==References== {{reflist}} * ''Appendix, the theory of space'' Janos Bolyai, 1987, p.143 {{Manifolds}} Category:3-manifolds Category:Curves Category:Hyperbolic geometry --- Adapted from the Wikipedia article [Horosphere](https://en.wikipedia.org/wiki/Horosphere) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Horosphere?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.