By David, 14 October 2021 Activity

**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!**

- 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)

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.

- Rule a 5 centimetre line coming in from the middle of the short side of your A4 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.
- 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.
- 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.
- 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.
- 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.
- Repeat steps 4–6 again twice to draw the final 2 sectors.

- Carefully cut out the template. Don’t fold anything!
- Bring the straight sides together so the shape makes a loop.
- Tape the straight sides together. Put tape on both sides of the cardboard to make this join strong.
- 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.
- Finally, tape together the 2 curves on the bottom of the ring. Your shape is now finished!
- 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!

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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