Moebius strip

A moebius strip is a one-sided surface that can be constructed by affixing the ends of a rectangular strip after first having given one of the ends a one-half twist. This space exhibits interesting properties, such as having only one side and remaining in one piece when split down the middle. The properties of the strip were discovered independently and almost simultaneously by two German mathematicians, August Ferdinand Moebius and Johann Benedict Listing, in 1858.

by M.C. Escher

A Moebius Strip is a twisted loop, normally made of paper.

Mathematical Idea

A twisted loop is very different from a normal loop.

Materials Needed

Strips of paper, sticky tape, scissors and a pen.

Demonstration

Take a strip of paper and some sticky tape. Turn the paper into a loop, but before you stick it down, flip one end of the paper over. This should give you a piece of paper with a half-twist in it. This is a Moebius strip. (see image.)

The Moebius strip has several strange properties.

The Moebius strip has only got one side. If you draw a line down the middle of the strip until you get back to your starting point, you will find that you draw on both sides of the paper. The twist in the paper makes you change sides as you draw around.

The Moebius strip used to be common in belt drives (like a car fan belt). With an ordinary belt only the inside of the belt was in contact with the wheels, so it would wear out before the outside did. Since a Moebius strip has only one side, the wear and tear on the belt was spread out more evenly and they would last longer. However, modern belts are made from several layers of different materials, with a definite inside and outside, and do not have a twist.

Similarly, the Moebius strip has only one edge. Make a mark on one point of the edge. Now start at the mark and trace along the edge with your finger. You will find that you get to the opposite point on the edge before your get back to the starting point.

Cut down the middle of the strip. Instead of getting two separate strips, the Moebius strip becomes one long strip. (To start the cut off, fold the strip and make a small cut, then unfold the strip and use the hole as a starting point.)

This long strip has four half-twists in it. If you cut it down the middle, you get two strips wound around each other.

Using a new strip, cut about a third of the way in from the edge (you will need to go around the loop twice). You will get two strips. One is a thinner the Moebius strip, the other is a long strip with four twists in it.

There are other combinations of strips that give interesting results when you cut down the middle of them:

Double Twist

Just like making a Moebius Strip, but this time flip the strip twice before sticking it down. This will give you a strip with two half-twists in it. This strip has two sides and two edges (check). When you cut down the middle of the strip it will split into two linked rings.

Triple Twist

Make a strip with three half-twists. This strip will have only one side and one edge. When you cut down the middle, the strip will become a knotted loop.

You can also stick two strips together to get some interesting effects. (See illustration)

Take two ordinary, untwisted loops. Stick them together at right angle to each other.

Now do the same with an ordinary strip and a Möbius strip.

With each set of strips, if you cut down the middle of both strips, you will get a square.

How did two different shapes both end up making a square?

Try doing it again, but this time colour in one side of the untwisted strip in both sets.

Can you see a difference in the squares now?

Hearts

Make two Moebius strips. One strip has to be twisted clockwise, the other anticlockwise. Stick them at right angles as before.

When you cut down the middle of both strips, you will get a pair of hearts linked together.

If you get a single heart and a separate, severely twisted heart, you didn't have the strips twisted in opposite directions.

Further Activities

Make the different strips mentioned above.

Try one half twist, two half-twists, three half-twists etc. Is there a pattern to the shapes you get?

Even numbers of half-twists give two loops, odd gives one loop. There is also a pattern to the number of times they twist around each other.

What do you get with more than two strips stuck together?

. A strip does not need to be twisted only once for it to be considered a moebius strip, in fact if you twist it 3 times, 5 times, or any odd number of times you will create a moebius strip.

TOPOLOGY

 Topologically the two objects are the same if they can be stretched into one another, as if they were made of play-doh (cutting and pasting are not allowed).

That’s why the loop and the moebius strip are different. One cannot be stretched into the other because they have different numbers of faces.

Spaces that can be stretched into one another share common rules of addition and multiplication (and other types of math).

The Alphabet: which letters of the alphabet are topologically equivalent?

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

This has to do with the properties of each letter: how many vertices, edges, faces, and holes it has.