Why Did the Chicken Cross the Möbius Strip?
A very good question you may be thinking (I'd hope so).
First let's start with the basics – What exactly is a Möbius strip when it's at home. Well it's a very special shape that has only one surface. It is in fact only a surface, and not really a shape at all, but I'm not really too fussed about the technicalities. Möbius strips are, contrary to what you may first think, very simple to make (Their big brother, the klein flask, is another kettle of fish altogether though), all you need is a strip of paper and some sellotape (If you're lacking the latter, ask Dave, he often has some in his bag). All you have to do is put a 180o twist in the strip of paper, and then sellotape the two ends together. You should end up with something looking a bit like this:
Now for the bit where you get to prove that your Möbius strip is an actual Möbius strip. You'll need to get a pencil or pen, or if none of these is available (and you've asked the ever useful Dave) your finger (which I hope is available) and draw a line along one surface (the only one) of the strip from the sellotape, all the way round until you've got back to the tape. Hopefully you've realised that there is only one surface (This is why a pen or pencil is advantageous – you can see where you've been)
You may be wondering what to do with your Möbius strip now you've made it, and you may not be too pleased to hear that I recommend cutting it up, as you get some quite interesting things. The first cut I'd recommend would be to split it down the middle (ie. cut along a line that is parallel to and equidistance from both edges). You will get a single strip which forms a figure of eight, but is not a Möbius strip. This may be interesting enough for some of you (Many expect it to form two seperate mini-Möbius strips), but if it isn't then I'd recomend repeating the operation (ie. cutting your new figure of eight down the middle), and you'll end up with two thin strips that are wound around each other. If you get a fresh Möbius strip and instead of splitting it down the middle, cut it about about a third of the way in (ie. cut along a line that is parallel to both edges and twice the distance from one edge as the other). You'll end up with a new Möbius strip and a loop with two half half twists in it (Not a Möbius strip). An apparently well known limrick is as follows:
- "A mathematician confided
- That a Möbius band is one-sided,
- And you'll get quite a laugh,
- If you cut one in half,
- For it stays in one piece when divided"
There are more proper uses for these special strips though, such as on giant conveyor belts, as both sides wear at the same rate meaning the belt does not have to be turned when one side starts to wear thin. A Möbius resistor is an electrical component that cancels its own inductive resistance (As an aside, Nikola Tesla, yes that Tesla, the one who also invented this beast, also patented a similar device).
William Hazzlett Upson wrote the short story "A. Botts and the Möbius Strip" in 1945, which was first published in the Saturday Evening Post and is about a worker who secretly restitches a conveyor belt to prevent his supervisor from being able to carry out his instruction to paint the outside of the belt and not the inside as a safety measure (I don't really understand how not painting the inside improved safety, but anywho)
Arthur C Clarke's "The Wall of Darkness" is an example of a Sci-Fi book that suggests that our universe is a generalised Möbius strip, and A J Deutsch's "A Subway Named Möbius" describes a time when the Boston becomes so complex that it forms a Möbius strip and trains start to disappear (Crazy American tube designers!)
M C Escher is one artist who likes the Möbius strip, and it fetaures in many of his artwork, such as this piece entitled "The Möbius Strip II" (I haven't yet found any reference to "The Möbius Strip I"):
The reason the chicken crossed the Möbius strip then, was to get to the same side. This simple joke has been described to me by a friend as a pretty good joke, for a geek joke, especially compared to the best mathematical joke she knew; What did the zero say to the eight…nice belt. I can see that one being even more dificult to write about, but Dave, Steve or myself will no doubt try some time in the future. Any other maths jokes that I could write about would be appreciated
This has been only a very basic overview of what a Möbius strip is, but i hope to expand this in the future.
Also, I feel that it is only fair that I acknowledge the alimighty wikipedia for the images and some other bits