If you have never heard of it before this
mathematical curiosity will blow your mind.To be quite honest though, maybe it only
blew MY mind because of my lamentable lack of spatial awareness. Being a typical girl, I cannot parallel park
and I cannot turn a cartwheel. (Though
now in my 50s I am a rather elderly girl and have pretty much given up on that one). I must also admit that, as a student, I was
no mathematical genius. Always more of a
words than a numbers person, I only managed to scrape a reasonable GCSE grade
thanks to the unbelievable patience and remedial efforts of my lovely maths
teacher Miss Williams. Nearly 40 years ago she had the dubious honour of trying
to instill a basic grasp of the rudiments of arithmetic, algebra, and geometry
into my thick skull, and she is still receiving therapy for that today. That
poor woman deserves a medal.
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| Mobius Strip with Ants, by Escher |
Nonetheless, although I am no
mathematician, I can safely say I am a true Geek. That is to say, I am curious
about the world around me and take a keen interest in all sorts of subjects,
whether or not I am any good at them. I was reminded of this intriguing phenomenon just last
week when the cold English winter drove me out to the shops to look for a cosy winter
scarf, and I found one in the form of a Mobius Strip. The paradox of the Mobius Strip made a big impression
on me when I first learned about it, and I hope you will find it interesting
too.
Sometimes also known as the Mobius band, the
Mobius Strip was discovered in 1858 by August Ferdinand Möbius and also
(coincidentally and independently) by Johann Benedict Listing, both German
mathematicians. Like many of the best ideas, the basic concept is simple, and
childishly easy to demonstrate. By the
way, for anyone with young kids, this is a good way to get them interested in
mathematics. It’s never too early to
start training a Geek!
All you need to do this is some paper and glue
or sticky tape. Start by making a long paper strip: it doesn’t particularly matter how long or how wide you make
it, but say 5 cm wide by 30 cm long, for example. Now take your glue or a bit of sticky tape.
You are going to stick the two ends together, but before you do that, just put
a half-twist in the band of paper, et voila! You have a Mobius strip.
By introducing that simple half-twist you
will find that your two-sided strip of paper has now miraculously become a
single-sided strip. It is no longer
possible to take two coloured pencils and colour the strip differently on
either side. Don’t believe me? Just try
it. Now you have made your Mobius strip, you can start playing.
Some Mobius Strip Experiments
Now take a pair of scissors and cut along the middle of your Mobius strip lengthways, along the line you have just drawn. You will find that is stays in one piece, twice as long as the original.
Draw a new line on your strip, about a third along its width. Once again, you will find the line eventually joins up to the point you started from.
Now cut along your newly drawn line. What do you think is going to happen? I bet you didn’t guess that you would end up with two linked loops!
Mobius Strips in Literature
If you enjoy reading classic American authors and
humorists like Mark Twain, Damon Runyon and Bill Bryson you should check out William
Hazlett Upson. Back in the 1940s and 1950s
he wrote a series of short stories for
the Saturday Evening Post newspaper about Alexander Botts, a salesman for the
Earthworm Tractor Company. In A Botts
and the Mobius Strip the eponymous hero thwarts his boss by restitching a
factory conveyor belt into the form of a Mobius Strip, thus preventing the
outside of the belt from being painted a different colour from the inside. TBH,
I have not actually read this yet. I got
as far as visiting the Amazon store, and found a copy of the Alexander Botts
Earthwork Tractors collection priced at £1,539.65. I was naively hoping for a free download for
my Kindle, but no such luck!
Moving to the world of Sci-Fi, in The Wall of Darkness by Arthur C Clarke,
the universe is re-imagined as a Mobius Strip, and in A Subway Named Mobius by AJ Deutsch the Boston underground network
becomes so huge and complex that the
system starts to behave according to some mysterious mathematical principles, causing trains to disappear.
Also worthy of mention in this context are John Barth’s Lost in the Funhouse, Vladimir Nabokov's
The Gift, and the movie Donnie Darko.
Finally, a Mobius Strip Joke
Why did the chicken cross the Mobius Strip? To get to the same side, of course.
FootNote: Some
Related phenomena for you to Google and introduce in your geeky maths lessons:-
Cross cap
Strange Loop
Klein bottle
Paradox
