A Möbius strip.
Cut a 5 cm strip lengthwise from paper (an old newspaper
will do). Holding the strip out straight, give one end a half twist (180º) and
glue or tape the two ends together. Your piece of paper is now a Möbius strip.
When you twisted your strip, the inside and the outside became one continuous
surface. There is also only one edge.
Take a pen and
carefully draw a line along the centre of a new uncut strip. Where do you end
up? Is the line drawn on the inside or outside of the paper? Now cut the strip
along the line you drew. How many pieces do you get? It may help if you use the
picture below to make an ant-covered Möbius strip.
Blow the above image up on a
photocopier, so the chain of ants is 23 cm long then join two copies, as shown
below, and do back-to-back photocopies. You need to experiment to get the ants
on opposite sides of the page, going in opposite directions. I had a bit of
trouble following my own instructions, so here’s a step-by-step set of photos:
(1) PDF on-screen; (2) printed out; (3) cut
up; (4) trimmed; (5) joined; and (6) a finished Möbius strip.
Next, take the photocopied or printed sheets and cut two
strips, 23 cm x 7 cm, and join them, so all the ants are in columns, and make a
Möbius strip which you can cut, either straight down the centre (see left), or
off to one side, as shown in the right-hand picture.
Try this again.
But this time, give the paper a full twist. Then try one and a half twists, and
see what happens. Last of all, see what you can discover about Klein bottles.
Some notes:
The pictures below
show what you get when you cut the strip. The first picture shows that a cut
down the middle gives a single loop, but there is a surprising result when you
test for Möbiusness (my own word). The test is simple: draw a pen line along
one side until you get back to the start: If the paper is still a Möbius strip,
the line will be on both sides, but in the first picture, that doesn’t happen:
Now in the second
picture, there are two interlinked loops. I cut off the big ants, and something
odd happened: the little ants are isolated on a Möbius strip, but the big ants
are on a non-Möbius strip.
Another Möbius strip, but this one has been
cut
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