Editing Möbius strip (section)

===Making the boundary circular===
{{multiple image|total_width=480
|image1=Mobius to Klein.gif|caption1=Gluing two Möbius strips to form a Klein bottle
|image2=MobiusStrip-02.png|caption2=A projection of the Sudanese Möbius strip}}
The edge, or [[boundary (topology)|boundary]], of a Möbius strip is [[homeomorphic|topologically equivalent]] to a [[circle]]. In common forms of the Möbius strip, it has a different shape from a circle, but it is [[unknot]]ted, and therefore the whole strip can be stretched without crossing itself to make the edge perfectly {{nowrap|circular.{{r|hilbert-cohn-vossen}}}} One such example is based on the topology of the [[Klein bottle]], a one-sided surface with no boundary that cannot be embedded into three-dimensional space, but can be [[Immersion (mathematics)|immersed]] (allowing the surface to cross itself in certain restricted ways). A Klein bottle is the surface that results when two Möbius strips are glued together edge-to-edge, and{{snd}}reversing that process{{snd}}a Klein bottle can be sliced along a carefully chosen cut to produce two Möbius {{nowrap|strips.{{r|spivak}}}} For a form of the Klein bottle known as Lawson's Klein bottle, the curve along which it is sliced can be made circular, resulting in Möbius strips with circular {{nowrap|edges.{{r|ddg}}}}

Lawson's Klein bottle is a self-crossing [[minimal surface]] in the [[unit hypersphere]] of 4-dimensional space, the set of points of the form <math display=block>(\cos\theta\cos\phi,\sin\theta\cos\phi,\cos2\theta\sin\phi,\sin2\theta\sin \phi)</math> for {{nowrap|<math>0\le\theta<\pi,0\le\phi<2\pi</math>.{{r|lawson}}}} Half of this Klein bottle, the subset with <math>0\le\phi<\pi</math>, gives a Möbius strip embedded in the hypersphere as a minimal surface with a [[great circle]] as its {{nowrap|boundary.{{r|schleimer-segerman}}}} This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the {{nowrap|1970s.{{r|sudanese}}}} Geometrically Lawson's Klein bottle can be constructed by sweeping a great circle through a great-circular motion in the 3-sphere, and the Sudanese Möbius strip is obtained by sweeping a semicircle instead of a circle, or equivalently by slicing the Klein bottle along a circle that is perpendicular to all of the swept {{nowrap|circles.{{r|ddg|franzoni}}}} [[Stereographic projection]] transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its {{nowrap|boundary.{{r|ddg}}}} The most symmetric projection is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles, but produces an unbounded embedding with the projection point removed from its {{nowrap|centerline.{{r|schleimer-segerman}}}} Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the [[orthogonal group]] {{nowrap|<math>\mathrm{O}(2)</math>,}} the group of symmetries of a {{nowrap|circle.{{r|lawson}}}}

[[File:Cross-cap level sets.svg|thumb|upright=0.8|Schematic depiction of a cross-cap with an open bottom, showing its [[level set]]s. This surface crosses itself along the vertical line segment.]]
The Sudanese Möbius strip extends on all sides of its boundary circle, unavoidably if the surface is to avoid crossing itself. Another form of the Möbius strip, called the '''cross-cap''' or '''crosscap''', also has a circular boundary, but otherwise stays on only one side of the plane of this {{nowrap|circle,{{r|huggett-jordan}}}} making it more convenient for attaching onto circular holes in other surfaces. In order to do so, it crosses itself. It can be formed by removing a [[quadrilateral]] from the top of a hemisphere, orienting the edges of the quadrilateral in alternating directions, and then gluing opposite pairs of these edges consistently with this {{nowrap|orientation.{{r|flapan}}}} The two parts of the surface formed by the two glued pairs of edges cross each other with a [[Pinch point (mathematics)|pinch point]] like that of a [[Whitney umbrella]] at each end of the crossing {{nowrap|segment,{{r|richeson}}}} the same topological structure seen in Plücker's {{nowrap|conoid.{{r|francis}}}}