the möbius strip

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The Möbius Strip

The Möbius strip or Möbius band (alternatively written Mobius or Moebius in

English) is a surface with only one side and only one boundary component. The

Möbius strip has the mathematical property of being non-orientable. It was

discovered independently by the German mathematicians August Ferdinand Möbiusand Johann Benedict Listing in 1858.

A model can easily be created by taking a paper strip and giving it a half-twist, and

then joining the ends of the strip together to form a loop. In Euclidean space there

are in fact two types of Möbius strips depending on the direction of the half-twist:

clockwise and counterclockwise. The Möbius strip is therefore chiral , which is to say

that it has "handedness" (right-handed or left-handed).

Properties The Möbius strip has several curious properties. A line drawn starting from the

seam down the middle will meet back at the seam but at the "other side". Ifcontinued the line will meet the starting point and will be double the length of the

original strip. This single continuous curve demonstrates that the Möbius strip has

only one boundary.

Cutting a Möbius strip along the center line yields one long strip with two full twists

in it, rather than two separate strips; the result is not a Möbius strip. This happens

because the original strip only has one edge which is twice as long as the original

strip. Cutting creates a second independent edge, half of which was on each side of

the scissors. Cutting this new, longer, strip down the middle creates two strips

wound around each other, each with two full twists.

If the strip is cut along about a third of the way in from the edge, it creates twostrips: One is a thinner Möbius strip— it is the center third of the original strip,

comprising 1/3 of the width and the same length as the original strip. The other is a

longer but thin strip with two full twists in it— this is a neighborhood of the edge of

the original strip, and it comprises 1/3 of the width and twice the length of the

original strip.

Other analogous strips can be obtained by similarly joining strips with two or more

half-twists in them instead of one. For example, a strip with three half-twists, when

divided lengthwise, becomes a strip tied in a trefoil knot. (If this knot is unravelled,

the strip is made with eight half-twists in addition to an overhand knot.) The

equation for the numbers of twists after cutting a Mobius strip is 2N+2=M, where N

is the number of twists before and M, the number after. Cutting a Möbius strip,giving it extra twists, and reconnecting the ends produces figures called paradromic

rings.

A strip with an odd-number of half-twists, such as the Möbius strip, will have only

one surface and one boundary. A strip twisted an even number of times will have

two surfaces and two boundaries.



If a strip with an odd number of half-twists is cut in half along its length, it will result

in a longer strip, with the same number of loops as there are half-twists in the

original. Alternatively, if a strip with an even number of half-twists is cut in half

along its length, it will result in two conjoined strips, each with the same number of

twists as the original.

Occurrence and use in nature and technology There have been several technical applications for the Möbius strip. Giant Möbius

strips have been used as conveyor belts that last longer because the entire surface

area of the belt gets the same amount of wear, and as continuous-loop recording

tapes (to double the playing time). Möbius strips are common in the manufacture of

fabric computer printer and typewriter ribbons, as they allow the ribbon to be twice

as wide as the print head while using both half-edges evenly.

A device called a Möbius resistor is an electronic circuit element which has the

property of canceling its own inductive reactance. Nikola Tesla patented similar

technology in the early 1900s: "Coil for Electro Magnets" was intended for use withhis system of global transmission of electricity without wires.



Taken from http://en.wikipedia.org/wiki/Möbius_strip

http://en.wikipedia.org/wiki/M%C3%B6bius_strip






The öbius Strip

There are two sides to everything. Right? A sphere has an inside and an outside. A

bug could be trapped inside, or it could crawl around on the outside. A piece of

paper sitting on a desk has a top and a bottom. Pages in a book are usually

numbered two per sheet of paper, with a front and a back. Can you think of anything that only has one side?

Follow the directions below, recording observations, predictions, and

outcomes as prompted.

To make a Mobius strip, cut out a strip of paper. Then take one side and give it a

half-twist. Next tape the ends together.

Journal entry #1: Compare and contrast this to a loop with no twist in it.

Draw a line along the center of the Mobius strip (parallel to the edges) until you get

back to where you started.

Journal entry #2: What do you notice about drawing this line?

Journal entry #3: Given what you know about cutting paper in half, what do you

think will happen to the strip if you cut along the line you drew?

Now cut the strip along that line. Write what you observe. Does it agree with your

predictions? Note specific differences.

Make another Mobius strip.

Journal entry #4: What do you think would happen if instead of cutting halfwaybetween edges, you cut 1/3 from the edge all the way around?

Cut your strip accordingly and record observations. Does it agree with your

predictions this time?

Cut another strip and twist one end twice (instead of just once) before attaching the

ends.

Journal entry #5: What do you think will be the same and different about this

double-twisted strip?

Draw a line down the center of this strip and stop when you get back to the

beginning. How many sides does this strip have? What do you expect to happen

when you cut along that line?

Your collection of journal entries from this activity should be about one page in

length.

Turn the journal in when you are finished.

Adapted from Kenneth Fuller, 2001



M.C. Escher Mobius Strip II

the möbius strip

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