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Impossible plait

By David, 23 September 2014 Activity

Cutting felt into strips

Fold a rectangle of felt in half and make two cuts coming in from the folded side. Stop cutting 2 cm before you reach the edge.

You will need

  • Felt sheet
  • Good scissors

Cutting the felt

  1. Cut a rectangle of felt about 6 cm wide and 20 cm long.
  2. Fold the felt in half so the short sides match up.
  3. Make two cuts, equally spaced, coming in from the folded side. Stop cutting about 2 cm before you reach the short edges of the rectangle.
  4. Unfold the felt. You will have a rectangle of felt with two long cuts in it. But it is more helpful to imagine the felt as three strips that are joined at the top and the bottom.
    Moving lefthand strip of felt over middle strip of felt

    Take the leftmost strip and bring it over one strip, so it is in the middle.

Making the plait

  1. Start at the top of the felt. There are three strips coming down from the top.
  2. Take the leftmost strip, and bring it over one strip to put it in between the two other strips.
  3. Now take the rightmost strip and bring it over one strip (the one you moved previously!) to put it in the middle.
  4. Take the new leftmost strip, and bring it to the middle. Then take the new rightmost strip and bring it over to the middle. There will be a bit of a tangle at the bottom of the felt – you will deal with that later.
  5. Repeat this pattern of bringing one strip from the left and one from the right one more time (making six movements in total). Then, hold the end of the plait with one hand.
    Felt is plaited at the top

    Repeat a pattern of bringing one strip from the right to the middle, and then one from the left to the middle.

  6. With your other hand, slowly begin to untangle everything below your grip. It can take a while, but eventually you should have three untwisted felt strips at the bottom, and your plait at the top.
  7. Release your grip, and then neaten up the plait.

What’s happening?

When you make your plait, you’re actually making two plaits at the same time. One plait is at the top of the felt, and the other is at the bottom. You can untangle the plait at the bottom without touching the plait at the top. But just as easily, you could untangle the plait at the top, and keep the plait at the bottom.

Untangling the felt strips

Hold the plait with one hand, and use the other hand to untangle the bottom part of the felt.

Your plait looks like a knot. But mathematically speaking, it’s not a knot!

This is an area of maths called knot theory. The most basic problems in knot theory look at what you can do with a rubber band. There are many shapes you can twist the band into. But to make some shapes, such as a trefoil knot, you first have to cut the rubber band.

In knot theory, two loops are the same if you can stretch one to make the other, and they are different if you need to break them to make the change. To make this plait, you did not need to rip the felt strips, so this is not a knot!

Neat plait

Release your grip and neaten up the plait.

More information

Make another impossible connection with zip ties

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