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Hyperplane
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a hyperplane is a generalization of a <a href="/facts/Plane_(geometry)/tAUtRFcn">two-dimensional plane</a> in <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a> to <a href="/facts/Space_(mathematics)/G6rt4bQ8">mathematical spaces</a> of arbitrary <a href="/facts/Dimension/0ktmUyac">dimension</a>. Like a <a href="/facts/Euclidean_planes_in_three-dimensional_space/UqRo1Bs0">plane in space</a>, a hyperplane is a <a href="/facts/Flat_space/vDkfkQJn">flat</a> <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a>, a <a href="/facts/Subspace_(mathematics)/G6rt4bQ8">subspace</a> whose <a href="/facts/Dimension/0ktmUyac">dimension</a> is one less than that of the <a href="/facts/Ambient_space/umgCRklf">ambient space</a>. Two lower-dimensional examples of hyperplanes are <a href="/facts/One-dimensional/bC4W2dHd">one-dimensional</a> <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a> in a plane and <a href="/facts/Zero-dimensional/ywL3gK7q">zero-dimensional</a> <a href="/facts/Point_(geometry)/6YPAvX2a">points</a> on a line. </p><p>Most commonly, the ambient space is n-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, in which case the hyperplanes are the (<i>n</i> − 1)-dimensional <a href="/facts/Flat_(geometry)/IlMjx7RK">"flats"</a>, each of which separates the space into two <a href="/facts/Half-space_(geometry)/AGknBgnw">half spaces</a>. A <a href="/facts/Reflection_(geometry)/5JHWnrql">reflection</a> across a hyperplane is a kind of <a href="/facts/Motion_(geometry)/x5zUzg7l">motion</a> (<a href="/facts/Geometric_transformation/qRjrasJf">geometric transformation</a> preserving <a href="/facts/Distance_(mathematics)/hbXlaEFk">distance</a> between points), and the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of all motions is <a href="/facts/Generating_set_of_a_group/B9q5lnFY">generated</a> by the reflections. A <a href="/facts/Convex_polytope/2pRHcRgC">convex polytope</a> is the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of half-spaces. </p><p>In <a href="/facts/Non-Euclidean_geometry/efKa7De2">non-Euclidean geometry</a>, the ambient space might be the <a href="/facts/N-sphere/bFu2S7E2">n-dimensional sphere</a> or <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic space</a>, or more generally a <a href="/facts/Pseudo-Riemannian_manifold/VFXsVjqb">pseudo-Riemannian</a> <a href="/facts/Space_form/m8jfVEI5">space form</a>, and the hyperplanes are the hypersurfaces consisting of all <a href="/facts/Geodesic/fbWR6l80">geodesics</a> through a point which are <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> to a specific <a href="/facts/Normal_(geometry)/jhtLIhcu">normal</a> geodesic. </p><p>In other kinds of ambient spaces, some properties from Euclidean space are no longer relevant. For example, in <a href="/facts/Affine_space/tlvI3oX1">affine space</a>, there is no concept of distance, so there are no reflections or motions. In a <a href="/facts/Orientability/1SEQM2Pt">non-orientable</a> space such as <a href="/facts/Elliptic_space/oMnFAoVb">elliptic space</a> or <a href="/facts/Projective_space/7MSpA9cM">projective space</a>, there is no concept of half-planes. In greatest generality, the notion of hyperplane is meaningful in any mathematical space in which the concept of the dimension of a <a href="/facts/Topological_subspace/NjU6z4Ca">subspace</a> is defined. </p><p>The difference in dimension between a subspace and its ambient space is known as its <i><a href="/facts/Codimension/AlGab3eJ">codimension</a></i>. A hyperplane has codimension 1. </p>