In mathematics, particularly hyperbolic geometry, a horoball is a specific kind of n-dimensional object in hyperbolic n-space. Consider the upper half-space model of hyperbolic space. Then a horosphere is given by a horizontal plane, and the horosphere and the space above it is called a horoball. Anything isometric to this is called a horosphere or horoball respectively. For n = 2 a horosphere is called a horocycle.

This terminology is due to William Thurston, who used it in his work on hyperbolic 3-manifolds. Thus horosphere/horoball often has a connotation of referring to 3-dimensional hyperbolic geometry.

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