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Zig-zag roller

By David, 14 October 2021 Activity

Two cardboard shapes held together with tape, one green, one white.

The sphericon has a strange way of rolling

This weird, wonderful shape is called a sphericon and it has a strange way of rolling. It wiggles back and forth as it trundles downhill!

You will need

  • A4 sheet of thin card
  • 10 centimetre wide protractor
  • Ruler
  • Pencil
  • Scissors
  • Sticky tape
  • Big, flat item that can be propped (e.g. cutting board, hardcover book)

Drawing the template

The template we’re drawing is made of 4 wide pizza slices, known as sectors. They will zig-zag across the page, one going up and the next down.

If you don’t want to draw the template, you can zigzag roller template and print it out on thin card.

  1. Drawing a line on some green card.Rule a 5 centimetre line coming in from the middle of the short side of your A4 card.
  2. Using a protractor to draw a curve onto the card.Measure a 127° angle from the end of the line, and also trace the outside of your protractor to make a curved arc to the 127° mark.
  3. Ruling an angle onto the card using the protractor.Rule a line from the centre of the angle to the end of the arc to make a sector. This line should also be 5 centimetres long.
  4. Ruling a line onto the card using the protractor.Put the protractor on the line you just drew, but put it upside down and with the centre of the protractor on the far end of the line.
  5. Using the protractor to draw another curve opposite to the first.Once again, measure a 127° angle from the end of the line, and also trace the outside of your protractor to make a curved arc to the 127° mark.
  6. Rule a line from the centre of the angle to the end of the arc to make a sector. This line should also be 5 centimetres long.
  7. Drawing multiple curves fitting together but opposite one another.Repeat steps 4–6 again twice to draw the final 2 sectors.

Making the roller

  1. Cutting out the curves with scissors.Carefully cut out the template. Don’t fold anything!
  2. Joining the ends to make a loop of card.Bring the straight sides together so the shape makes a loop.
  3. Taping the ends together.Tape the straight sides together. Put tape on both sides of the cardboard to make this join strong.
  4. Taping the top curves together.On top of the ring, there are 2 curves that need to be stuck together. Use a few little bits of tape (about 5 millimetres wide), to stick these curves together.
  5. Taping the lower curves together.Finally, tape together the 2 curves on the bottom of the ring. Your shape is now finished!
  6. The sphericon shape balanced on a book.Grab something big and flat, such as a cutting board or a hardcover book. Prop it up by a centimetre on one end, and then try to roll your shape down the slope. It will wobble back and forth as it goes!

 

What’s happening?

It might be soothing to watch a sphericon roll down a slope, but it’s also interesting for mathematicians to think about.

One reason a sphericon rolls so well is to do with its centre. As it rolls, the centre stays at the same height from the ground. That’s true for balls and wheels too. It’s not true when you try to roll a box shape. As a box comes up onto a corner, the middle of the box rises, and it drops again as the box settles onto a face.

The sphericon has two edges, and only one face. You can trace your finger around the face in one big loop. When a sphericon rolls down a slope, the bit that touches the ground traces this same path.

As a sphericon rolls, every part of its surface touches the ground. Most rolling shapes don’t do this. If you roll a can, the circles at the ends don’t touch the ground. Even a ball doesn’t do it. When a ball rolls downhill, only a thin circle touches the ground!

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