<story><title>Who Expected That? Extreme close-ups create a Klein Bottle.</title><url>http://www.thebigquestions.com/2012/02/23/who-expected-that/</url></story><parent_chain><item><author>grannyg00se</author><text>Is it common to refer to the Möbius strip as a 2D surface?  You can&apos;t uniquely identify each point on its surface using only two dimensions, so I wouldn&apos;t call it a 2D surface at all.  It lives in 3D, therefore it is a 3D surface, no?</text></item><item><author>wisty</author><text>Well, a Möbius strip is a 2D surface that can&apos;t fit in 2 dimensions. Knots are 1D, and won&apos;t fit in 2 dimensions.</text></item><item><author>grannyg00se</author><text>&quot;a Klein bottle, which is a two-dimensional surface that can’t be squeezed into three dimensions, but fits perfectly well in nine (or for that matter in four). &quot;&lt;p&gt;I&apos;m having trouble squeezing this concept into my mind.</text></item></parent_chain><comment><author>dfranke</author><text>&lt;i&gt;You can&apos;t uniquely identify each point on its surface using only two dimensions&lt;/i&gt;&lt;p&gt;Yes you can. Run a sharpie along the bottom edge of the paper so that it bleeds onto both sides of the paper. Now tape the ends of the strip together with a half twist to make a Möbius strip. Place a point anywhere on the strip. Draw a line through that point such that the line meets both edges of the strip at a right angle.  Measure the distance along that line between your point and the darkened edge; call this &lt;i&gt;x&lt;/i&gt;. Now, hold the Möbius strip in your left hand so that you&apos;re pinching it by the tape. With your right hand, run your finger along the strip, starting from the tape and moving to the right. Measure how far you have to move your finger in order to reach the line you drew; this could require up to two loops around the strip. Call this distance &lt;i&gt;y&lt;/i&gt;. The tuple (x,y) uniquely identifies your point.</text></comment>
<story><title>Who Expected That? Extreme close-ups create a Klein Bottle.</title><url>http://www.thebigquestions.com/2012/02/23/who-expected-that/</url></story><parent_chain><item><author>grannyg00se</author><text>Is it common to refer to the Möbius strip as a 2D surface?  You can&apos;t uniquely identify each point on its surface using only two dimensions, so I wouldn&apos;t call it a 2D surface at all.  It lives in 3D, therefore it is a 3D surface, no?</text></item><item><author>wisty</author><text>Well, a Möbius strip is a 2D surface that can&apos;t fit in 2 dimensions. Knots are 1D, and won&apos;t fit in 2 dimensions.</text></item><item><author>grannyg00se</author><text>&quot;a Klein bottle, which is a two-dimensional surface that can’t be squeezed into three dimensions, but fits perfectly well in nine (or for that matter in four). &quot;&lt;p&gt;I&apos;m having trouble squeezing this concept into my mind.</text></item></parent_chain><comment><author>bmm6o</author><text>It&apos;s commonly called a non-orientable 2 dimensional manifold.  Locally, it looks like a smooth deformation of the plane.  The non-orientable bit is what distinguishes it (globally) from something more like a cylinder.</text></comment>