Mobius Strip

Mobius Strip
Field: Topology
Image Created By: David Benbennick
Website: Wikipedia

Mobius Strip

A Mobius strip, also referred to as a Mobius band, is a bounded surface with only one side and one edge.

Basic Description

File:313px-MöbiusStripAsSquare.svg.png

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The Mobius strip is a non-orientable surface with a boundary, or edge. If we take a rectangular strip of paper, then make a half twist and join the ends we come up with a Mobius strip. If we were to draw a line through the center of the strip without lifting the pencil off the paper, we would come back to the starting point but on the "opposite" side of the paper. Logically, this is only possible if the surface has only one side and only one boundary, meaning that, while the Mobius strip appears to have two sides, it actually has one.

The Mobius strip can be represented as a fundamental polygon, as shown in Image 1. The fundamental polygon provides a way to represent the shapes as a simple polygon. Arrows and labels indicate which sides are glued together, and in what orientation. If the corresponding red edges are joined so that the arrows line up, the result will be a Mobius strip.

Being non orientable means that if perpendicular arrows were drawn in the surface, one pointing north and the other pointing west, moving them along specific paths in the shape would return them to the starting point as a mirror image of the position they were in when they began. As a result of its non orientability, a Mobius strip has only one side and one edge when modeled in 3 dimensions.

Because the Mobius strip has only one edge, the edge forms a continuous loop that bounds the surface. Since any closed loop that isn't a knot can be smoothly turned into a circle, and vice-versa, all simple closed loops are homeomorphic to the circle. Hence, somewhat surprisingly, the edge of the Mobius strip is a circle. It is a highly distorted circle, but a circle nonetheless.

Picture a 2 dimensional crab walking along the center line of the Mobius strip; whereas on an orientable surface like a sphere it would have to go just once around the surface to arrive at its starting point in the same orientation in which it left; on a Mobius strip it would have to go around the loop twice. Any strip with an odd number of twists will behave the same way as a Mobius strip, that is, it will have one edge and one side. On the other hand, any strip with an even number of twists will have two boundaries and sides.

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Because the Mobius strip has an edge, it is not technically a [[Topology Glossary#Manafold|2 manifold [...]

[Click to hide A More Mathematical Explanation]

Because the Mobius strip has an edge, it is not technically a 2 manifold, or surface, as manifolds have no edges. Nevertheless, The Mobius strip is rather surface-like, so is often referred to as a surface with a boundary, to distinguish it from true surfaces.

Non-orientability

Notice that, in the construction of the Mobius strip via the fundamental polygon, points farther on the right of the top edge of the polygon are merged with those farther to the left on the bottom edge. Each pair of these matched points represents (maps to) a single point on the actual Mobius strip as in Image 2.

As a result of this mapping, the Mobius strip is non-orientable. Here is an example that illustrates the concept of non-orientability and how it arises on the Mobius strip.

Image 3 depicts the Mobius strip represented via a fundamental polygon. On the edges of the polygon are two pairs of identified points. Let us suppose again that a 2 dimensional crab lives within the Mobius strip. In Image 4, the fiddler crab is shown walking Toward the top edge of the polygon. Notice that his right side is moving toward the blue point, while his left side is headed for the green point. The points on the edge that he is approaching are the same as the corresponding points on the opposite edge. Thus, once the crab reaches the points on the top edge, he will emerge from bottom edge of the polygon. Because the crab's right side went to the blue point, and his left to the green point, his right side will emerge from the blue dot on the bottom, while his left will emerge from the green dot on the bottom, as in Image 5. This matching has the effect that, after traveling around the Mobius strip, the crab's right side has flipped with his left.

Image 2. The fundamental polygon folded into a Mobius strip. representation of the Mobius strip with two points located ate the merging.

Image 3. The fundamental polygon representation of the Mobius strip with two pairs of identified opposite points, each representing the single, corresponding point in Image 2.

Image 4. A fiddler crab walking toward two points on a Mobius strip.

Image 5. The fiddler crab having past the two points. He emerges from the opposite side of the polygon with his right and left sides flipped.

Parametrization

Image 6. A parametric representation of a Mobius strip generated from the equations to the left.

Image 7. The u and v axes labeled in a Mobius strip.

The following set of equations parametrize a Mobius strip of width 1 centered at (0,0,0) and whose middle circle has a radius of 1:

$x(u,v)= \textstyle \left(1+ v \cos \frac{1}{2}u\right)\cos u$

$y(u,v)= \textstyle \left(1+ v\cos\frac{1}{2}u\right)\sin u$

$z(u,v)= \textstyle v\sin \frac{1}{2}u$

where $0 \le u < 2 \pi$ and $-\tfrac{1}{2}\le v \le \tfrac{1}{2}$.

In this parametrization $u$ runs around the strip, while $v$ moves from one edge to the other. For example, if the fiddler crab living within the Mobius strip were to construct axes withing his flat, Mobius strip universe, he would orient the v-axis along one of the radial lines, while the u-axis would run around the loop of the strip, as illustrated in Image 7. Once $u$ reaches 2π, the surface begins repeating itself.

[Click to reveal an explanation of the parametrization of the Mobius strip.]

[Click to hide an explanation of the parametrization of the Mobius strip.]

The parametrization of the Mobius strip can be understood as a manipulation of that of the annulus. Note that all of the figures below have been rotated sightly from vertical so that they can be viewed better.

Step 1

Image 8. The annulus created by the parametrization to the left.

One parametrization for the annulus is:

$x(u,v)= \textstyle \left(1+ v \right)\cos u$

$y(u,v)= \textstyle \left(1+ v \right)\sin u$

$z(u,v)= \textstyle 1$

where $0 \le u < 2 \pi$ and $-\tfrac{1}{2}\le v \le \tfrac{1}{2}$.

In the x and y components, 1 + v defines the radius of the shape; as v goes from $-\tfrac{1}{2}$ to $\tfrac{1}{2}$, the annulus's width is set to be from 1 + $\left (-\tfrac{1}{2} \right ) = \tfrac{1}{2}$ units away from the center to 1 + $\tfrac{1}{2} = \tfrac{3}{2}$ units. Hence the hole in the center. u revolves this region around the center, sweeping out every point that is between that lies $\tfrac{1}{2}$ to $\tfrac{3}{2}$ units from the center.

Step 2

Image 9. The figure created by the parametrization to the left.

Next we change the z parameter to v, yielding:

$x(u,v)= \textstyle \left(1+ v \right)\cos u$

$y(u,v)= \textstyle \left(1+ v \right)\sin u$

$z(u,v)= \textstyle v$

where $0 \le u < 2 \pi$ and $-\tfrac{1}{2}\le v \le \tfrac{1}{2}$, as before.

All remains the same, except now, rather than lying all in the same plane at z = 1, the figure has a height that goes from z = $-\tfrac{1}{2}$ to z = $\tfrac{1}{2}$, in accordance with the variable v. Because v simultaneously defines the radius and height, the lowest height is where the radius is the smallest, and the greatest height is where the radius is largest. This gives the figure its conical shape.

Step 3

Image 10. The figure created by the parametrization to the left; it has one full twist in it.

Now, we Add a full twist to the figure as it loops, using the following equations:

$x(u,v)= \textstyle \left(1+ v \cos u \right)\cos u$

$y(u,v)= \textstyle \left(1+ v \cos u \right)\sin u$

$z(u,v)= \textstyle v \sin u$

where $0 \le u < 2 \pi$ and $-\tfrac{1}{2}\le v \le \tfrac{1}{2}$, as before.

Because v defines the radius of the figure, multiplying v by Cosu in the parametrizations for x and y, adds a rotational aspect to the radius. This is completed by multiplying v by Sinu in the parametrization for z. The difference between the equations for x and y versus z, in this step has to do with how the shape fits in space.

Image 11. A labeled cross-section of the surface.

The xy-plane intersects the figure in Images 9 and 10 lengthwise through the surfaces' centers, much like cutting a ribbon down the middle. Image 11 depicts a cross-section of the surfaces in Image 10 at an arbitrary location, showing the xy-plane intersects the surface at its midpoint. v specifies each point along the radius, or width, of the surface, so the red point on the cross-section is some value of v away from the center.

The cross-section creates a right triangle with legs that are the length of the red point's position on the xy-plane, and its location in the z direction. Using trigonometry, We find that the Point's xy position is given by v cosu and its z location, by v sinu. This explains the difference between the parametrization for x and y versus that of z.

Similar to what occurred in step 2 with the variable v defining both the radius and height of the figure, u also now defines two aspects of the figure as it goes from 0 to 2π. u rotates the figure's width about its center creating the loop, and revolves the width about its own midpoint, giving the surface a twist as it loops around the center.

Step 4

Image 12. The Mobius strip created by the parametrization to the left; it has one half twist in it.

To obtain a parametrization for the Mobius strip, we need the surface to make a half twist about its midpoint, rather than a full twist as it does in Image 10. This is quickly remedied by multiplying u by $\tfrac{1}{2}$ in the places where it defines the twist (as opposed to the places where it defines the creation of the loop):

$x(u,v)= \textstyle \left(1+ v \cos \frac{1}{2}u\right)\cos u$

$y(u,v)= \textstyle \left(1+ v\cos\frac{1}{2}u\right)\sin u$

$z(u,v)= \textstyle v\sin \frac{1}{2}u$

where $0 \le u < 2 \pi$ and $-\tfrac{1}{2}\le v \le \tfrac{1}{2}$.

We have now reached the original equations shown in the Parametrization section, and demonstrated their generation from those that create an annulus.

Relationship to the Cross-cap

Image 12. A depiction of the Cross-cap. The vertical line is the region of self-intersection, and the gray ovals define the cross-section.

Both the Cross-cap and the Mobius strip by removing a disk from the Real Projective Plane. While this process can be reversed in the case of the Mobius strip, leading back to the Real Projective Plane, a region of self intersection prevents the process from being reversible in the case of the Cross-cap. As such, the Mobius strip can be turned into the Cross-cap, but the reverse cannot be done. This fact means that the Cross-cap is simply a model of a Mobius strip, rather than homeomorphic to the Mobius strip. This topic is discussed in depth under the more mathematical explanation section of the Cross-cap page.

Why It's Interesting

Mobius Strips are one-sided, so if you were to walk along the strip, you would end up on the reverse "side" from where you started, without crossing over the edge.

Also, as discussed in the More Mathematical Explanation section, the Mobius strip is non-orientable. Were a 2 dimensional animal living within the Mobius strip within the strip to walk along the strip, it would return to its starting point with its left and right sides flipped.

There are also many cool experiments one can try with home-made mobius strips:

Experiment 1: Draw a line through the center of the strip. We would have to go round the loop twice to get back to the starting point. This is a key feature of the Mobius strip because it's what describes it as a non orientable surface.

Experiment 2: Cut through the center line. In general, if we cut a rectangular strip of paper lengthwise through the middle from end to end, we would expect to get two strips. This is not the case with the Mobius strip. Instead of getting two strips, we get one long strip with two full twists in it. If we cut the strip again through the center line, we come up with two strips wound around each other.

Experiment 3: Cut through the line about one third from the edge. (note: we have to go twice round the loop), we get two separate strips, one of which is thinner, but of the same length as the original strip. The other will be a long strip whose length is twice that of the original strip.

Experiment 4: Cut once round the loop through the line about one third from the edge, then cut through the center line of the resultant thicker strip. We get three strips wound around each other, one in the middle and one on either side.

Applications

The universal recylcing sign is a Mobius strip

B.F.Goodrich Company manufactures Turnover Conveyor Belt System which has half twists in it to allow for equal wearing on both sides of the belt

Mobius strips have been used as conveyor belts because their one sided nature allows equal wearing of "opposite sides" of the belt. This makes the belts last longer. The strip has also been used in recording tapes to double the playing time without having to manually take out the tape and change the side playing. It is also used in numerous electronic appliances especially those which have resistors and superconductors.

An artist like M.C. Escher who had recurring mathematical themes in his works used Mobius strip in some of his works. Escher's Parade of Ants is a lithograph which shows ants moving on the strip. Martin Gardner uses the idea of the Mobius strip in his amusing short story "No-Sided Professor," which can be found in the book Fantasia Mathematica.

The universal recycling sign is a Mobius strip.

Teaching Materials

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References

• Youtube video - http://www.youtube.com/watch?v=BVsIAa2XNKc
• Parametrization Image - http://mathworld.wolfram.com/MoebiusStrip.html
  
• Conveyor belt - http://www.math.unh.edu/cgi-bin/generatePage.cgi?moebius
  
• Universal recycling sign - http://chemistry.about.com/od/healthsafety/ig/Laboratory-Safety-Signs/Recycling-Sign.htm

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