A strange shape with rotational symmetryHave you ever made a paper snowflake? Here’s a different twist on a classic craft activity.

You will need

  • Paper
  • Scissors
  • Sticky tape

What to do

  1. Cut out two large circles of paper. Don’t worry if they’re not perfect!
  2. Fold one circle in half.
  3. Cut along the fold, but stop when you get to the middle.
  4. Someone rolling paper into a cone.Take one side of the cut, and bring it to the fold on the other side of the circle to make a cone.
  5. Wrap the other side of the circle around the cone, and hold it in place with a small piece of tape.
  6. Someone cutting a stringe shape out of a cone of paper.Cut a wiggly shape out of the side of the cone. Make sure you cut through both layers of paper!
  7. a strange shpe with rotational symmetryRemove the tape and unfold the cone. Does the shape have any symmetry?
  8. Now fold the other circle into quarters.
  9. Cut to the middle of the circle along one of these folds.
  10. Someone has folded a circle into four quarters, and then cut along one of the lines to the centre.Bring the newly cut edge to the near fold.
  11. Wrap the remaining paper around three more times to make a much sharper cone than the first.
  12. A shape with lots of rotational symmetry.Cut along the edge as before and when you unfold it, your new shape will have even more symmetries.
  13. Experiment with more snowflakes. Tighter winding will give you more arms on your snowflake!

What’s happening?

Have you ever made a paper snowflake? Fold a piece of paper several times and then cut an interesting curve along the surface. When you unfold the paper, you’ll get a cool shape with lots of symmetry. It will look similar to the snowflake in this activity, but not quite the same.

Folds add a symmetry known as mirror symmetry. If you put a mirror along a fold line of your snowflake, it’ll look just like the original snowflake.

The twisted snowflake in this activity doesn’t have mirror symmetry. It has rotational symmetry. Rotate your twisted snowflake on a flat surface. If you turn it a half circle, it will look the same as when you started.

By adding more windings before cutting, you add more and more rotational symmetries.

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