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Secondary Magazine - Issue 58: Focus on


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 13 April 2010 by ncetm_administrator
Updated on 29 April 2010 by ncetm_administrator

 

Secondary Magazine Issue 58
 

Focus on...the Möbius Strip

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Wolfram Mathworld says that ‘the Möbius Strip’, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided, non-orientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The ‘strip’ bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).


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August Ferdinand Möbius (1790 – 1868) was a German mathematician and astronomer. His father, a dance teacher, died when August was three years old, and Möbius was raised and home schooled by his mother until he was 13. A detailed biography of August Möbius can be found at the MacTutor History of Mathematics archive.


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Plus Magazine offers this for that special mathematician in your life… If you know anyone who might appreciate a rather geeky Valentine tribute, try this. Make two Möbius strips, with one twisted clockwise and the other twisted anticlockwise. (It won't work unless you get this right.) Stick the two loops together, at right angles to each other. Send this to your inamorata, with instructions to cut along the middle of both loops. (When you get to the join, cut through both loops.) What you get is a mathematical Valentine!


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It’s unlikely to surprise you to find that the Möbius strip featured in the work of MC Escher.


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The properties of the Möbius strip have found a practical use in a number of industries. Wikipedia states that, “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.”


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The musically-minded of you might appreciate this video of Bach’s Canon 1 à 2 (from A Musical Offering) written on a Möbius strip.


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As with much of mathematics, the Möbius strip needs to be experienced rather than just read about. Get a piece of paper and make yourself one, then…

  • challenge a class to colour only one side
  • ask a class to predict what will happen when it is cut in half through a line parallel to the edge (and then repeat the question and the cutting in half on the resulting band)
  • ask what would happen if, instead of cutting in half, the strip is cut following a line parallel to the edge but a third of the way across.

What happens if extra twists are included? Or if the cuts are made different fractions of the way across?


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Do you ever feel that your breakfast is lacking mathematics? Follow these instructions to cut your morning bagel so that the cutting surface is a two-twist Mobius strip; it has two sides, one for each half!
 
 
 
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