Klein bottle

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Klein bottle  is an example of a <a href="/facts/Orientability/1SEQM2Pt">non-orientable</a> <a href="/facts/Surface_(topology)/jK1b1KIG">surface</a>; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a <a href="/facts/Two-dimensional/f4b397W8">two-dimensional</a> <a href="/facts/Manifold/rXPERJtu">manifold</a> on which one cannot define a <a href="/facts/Normal_vector/jhtLIhcu">normal vector</a> at each point that varies <a href="/facts/Continuous_function/sKbl02pB">continuously</a> over the whole manifold. Other related non-orientable surfaces include the <a href="/facts/M%25C3%25B6bius_strip/35kzqqFL">Möbius strip</a> and the <a href="/facts/Real_projective_plane/gGVd0ctY">real projective plane</a>. While a Möbius strip is a surface with a <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a>, a Klein bottle has no boundary. For comparison, a <a href="/facts/Sphere/jywYLNM2">sphere</a> is an orientable surface with no boundary.
</p><p>The Klein bottle was first described in 1882 by the mathematician <a href="/facts/Felix_Klein/WmL9guqK">Felix Klein</a>.
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Klein bottle open-in-new

Klein bottle