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How Many Surfaces? Making a Möbius Strip

By: Suzanne Elvidge BSc (hons), MSc - Updated: 24 Aug 2012 |
 
Möbius Strip Paper Twist Geometry

A Möbius strip is a loop of paper with a twist in it, that behaves in a very odd way.

How to Make a Möbius Strip

Take a strip of paper, curve it round into a loop and glue the ends together. Draw a line all the way around the strip without taking the pencil off the paper. How many sides has it got? Run the pencil along the edge of the strip – how many edges has it got?

The simple loop has two surfaces and two edges.

Take a second strip of paper and curve it round into a loop, but before gluing the ends together, twist one end of the paper, to make a loop with a single twist in it. This is a Möbius strip. Draw a line all the way around the strip without taking the pencil off the paper. How many sides has it got? Run the pencil along the edge of the strip – how many edges has it got? Try making one with two twists in it – does that make any difference?

Because the Möbius strip has one twist in it, this connects the top surface of one end of the piece of paper to the bottom surface of the other end of the piece of paper. This creates a loop of paper with only one surface and one edge. When a strip has an even number of twists in it, this connects the top surface of one end with the top surface of the other, and so creates a loop with two surfaces.

Cut a Möbius strip in half, down the pencil line and describe what kind of strip is created. Is this a Möbius strip? Test by drawing a line down the centre. What happens when it's cut in half again?

This creates a one larger loop with two twists. Cutting it in half again makes two interlocking loops. Cutting each of these loops again will make four interlocking loops. Try it with three or more twists.

Try cutting a Möbius strip into a third instead of a half – is the result any different?

This creates two interlocking loops, one half the size of the other.

Who was Möbius?

The loop of paper with a twist was first described by a German mathematician called Johann Benedict Listing. August Ferdinand Möbius, who the strip is named after, was the son of a dancing teacher, and was born in November 1790. He was a German mathematician and astronomer, and he was interested in a type of geometry (the maths of points, lines, angles, surfaces, and solids) called 'topology'.

Uses for a Möbius Strip

Because a Möbius strip has only one side, it has a lot of possible uses in industry. A conveyor belt, sanding belt or drive belt made from a Möbius strip would wear evenly on both sides. A computer printer ribbon made from a Möbius strip would last twice as long, and a recording audio tape would record for twice as long.

A Möbius strip scarf will lie flat under a jacket or coat. The Möbius strip is used as the 'reduce – reuse – recycle' recycling symbol.

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