Mobius strip - Revision history

Vipul: 3 revisions

2008-05-18T19:49:56Z

3 revisions

← Older revision	Revision as of 19:49, 18 May 2008
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Vipul at 09:22, 1 August 2007

2007-08-01T09:22:22Z

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 Then the Mobius strip is the trace of a moving open line segment of length twice the half-width whose center traces the midcircle, and which rotates at a rate of half that at which it revolves.
 ==Equational descriptions==
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 <math>(x,1) \sim (-x,-1), (1,y) \sim (-1,y)</math>

Vipul: /* Topological structure */

2007-08-01T08:47:42Z

Topological structure

← Older revision		Revision as of 08:47, 1 August 2007
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	Topologically, the Mobius strip can be viewed as:		Topologically, the Mobius strip can be viewed as:

	<math>\{ (x,y} \| -1 \le y \le 1 \}</math>		<math>\{ (x,y) \| -1 \le y \le 1 \}</math>

	modulo the equivalence relation:		modulo the equivalence relation:

	<math>(x,1) \sim (-x,-1)</math>		<math>(x,1) \sim (-x,-1)</math>


	==Related surfaces==		==Related surfaces==

Vipul at 08:30, 1 August 2007

2007-08-01T08:30:20Z

New page

==Definition==

The '''Mobius strip''', or ''twisted cylinder'', is defined as follows.

* There is a ''midcircle'' of radius <math>R</math>
* There is a ''half-width'' <math>w</math>

Then the Mobius strip is the trace of a moving open line segment of length twice the half-width whose center traces the midcircle, and which rotates at a rate of half that at which it revolves.

==Equational descriptions==

===Cartesian parametric equation===

The Cartesian equation is given as follows, where <math>t</math> varies in <math>[0,2\pi)</math> and <math>s \in (-w,w)</math>:

* <math>x = (R + s \cos (t/2)) \cos t</math>
* <math>y = (R + s \cos (t/2)) \sin t</math>
* <math>z = R \sin (t/2)</math>

==Abstract structure==

===Topological structure===

Topologically, the Mobius strip can be viewed as:

<math>\{ (x,y} | -1 \le y \le 1 \}</math>

modulo the equivalence relation:

<math>(x,1) \sim (-x,-1)</math>

==Related surfaces==

{{one-point compactification|projective plane}}

The Mobius strip can be identified with the set of all lines in the [[affine plane]], and hence, its one-point compatification is that set plus the line at infinity, which is the projective plane.

{{a compatification|Klein bottle}}

One approach to compactifying the Mobius strip is to identify the two bounding sides, which, locally, looks like identifying the endpoints of each line segment. This gives rise to the [[Klein bottle]].

The Klein bottle, however, cannot be embedded in <math>\R^3</math>. It can be described topologically as follws:

<math>\{ (x,y) | -1 \le x \le 1, -1 \le y \le 1 \}</math>

modulo the following equivalence relations:

<math>(x,1) \sim (-x,-1), (1,y) \sim (-1,y)</math>

==External links==

* {{mathworld|MoebiusStrip}}