Möbius Strips

Cymraeg

Möbius Strips

This activity is designed to challenge your knowledge of shapes, whilst being fun and surprising.

As these exercises require scissors, we advise younger learners to be supervised by an adult.

Equipment Needed

This activity requires a sheet of A4 paper/card, scissors, pencil, and a glue stick.

You will need to cut your A4 paper/card into several 2cm wide strips. The orientation of the paper/card is unimportant, as the length of the strips will make no difference to the outcomes.

The Challenges

It is best to work through the below challenges in order.

Challenge 1

  1. Take one of your strips of paper (see the above Equipment Needed section for dimensions)
  2. Use some glue to connect the two ends in a loop.
  3. On the outside of the loop, draw a line around the middle.

    A strip of card connected into a loop with a dashed line drawn around the centre
  4. Predict what will happen if you cut the loop along this line.
  5. Test your prediction by cutting carefully along the line.

    You should now have 2 skinnier loops.

Challenge 2

  1. Take one of your strips of paper.
  2. This time, put a single twist in the strip before gluing the ends together. It should look like this:

    A strip of card with its ends taped together to form a loop with a single twist in it
  3. What happens when you try to draw a central line around the outside?

    4. This shape is called a Möbius Strip. It only has one surface.

  4. What do you think will happen if you cut along this line?
  5. Now test your prediction by cutting carefully along the line.


    6. A larger loop with several twists
  6. How many twists are there in this new shape? If unsure, you could cut the shape open and see how many twists are needed to straighten it out again.

    7. There are 4 twists in this new shape.

Challenge 3

  1. Use one of your strips of paper to create another Möbius Strip.
  2. This time, instead of trying to draw a line around the middle of this shape, draw it a third of the way from the edge.

    A möbius Strip with a line drawn around one third of the way from the edge.
  3. What do you think will happen if we cut along this new line?
  4. Now test your prediction by carefully cutting along the line.


    5. A Möbius Strip interlocked with a loop containing four twists.

    We now have a Möbius Strip connected to the same 4-twist loop we created in Challenge 2.

    Is this what you expected?

Challenge 4

  1. Get two new strips of card or paper of the same size and glue them together in a cross.

    Two strips of card glued in their centres to form a cross
  2. Tape one of these strips into a simple loop, the other as a Möbius Strip.

    Connected loop and Möbius Strip
  3. Draw a line around middle of each shape.
  4. Cut along the central line of both, not including the joining section.

    The shapes cut as described above
  5. Before we complete the cuts across the glued section, think about what shape(s) you think this action will create. We already know from the previous challenge what happened with just a Möbius Strip, and we know that a single simple loop cut along the centre line will produce two loops.
  6. When you're ready to check your predictions/thoughts, complete the cuts through the last section - you will need to hold them in place between the two cuts.


    7. A square formed by following this challenge's instructions.

    We now have a square!

    Is this what you expected?

Challenge 5

  1. Get two new strips of card or paper of the same size and glue them together in a cross.

    Two strips of card glued in their centres to form a cross
  2. This time we will tape them both as Möbius Strips with opposite twists. This means one will be twisted clockwise before taping, the other anti-clockwise.

    Connected Mobius Strips that have opposite twists
  3. Draw a line around middle of each shape.
  4. Cut along the central line of both, not including the joining section.

    The shapes cut as described above
  5. Before we complete both the central cuts through the joining section, let's consider what we think might happen this time.
  6. Once you are ready, complete the cuts to see how accurate your predictions were.


    7. Two interlinked hearts produced by following this challenge's instructions.

    We now have two interlocked hearts

    Tip: If you ended up with two separated shapes at this point it means the two Möbius Strips were twisted in the same direction, not opposite.