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There’s a hidden shape inside your star waiting to be discovered

The holidays are coming, and it’s time to start decorating. What better way than a bright, colourful, mathematical star decoration!

You will need

• Saucer
• Coloured paper
• Sharpened pencil
• Scissors
• White paper
• Ruler
• Protractor
• Sticky tape

What to do

1. Use the saucer and pencil to trace out six circles on coloured paper.
2. Cut out the circles using scissors.
3. Fold the circles in half and then cut along the folds so you get 12 semicircles.
4. Rule a 15-centimetre line on the white paper and mark the midpoint along the line.
5. From this midpoint, use your protractor to measure out lines at angles of 36°, 72°, 108° and 144°.
6. Place each semicircle on the white paper, centred at the midpoint. Mark the 36°, 72°, 108° and 144° angles around the edge of the semicircle.
7. Carefully bend each semicircle into a cone shape, with the marks facing out. When you’re happy with the shape, stick it together with sticky tape so the edges just meet.
8. Look at your cones. If you imagine the seam where you stuck them together as a mark, they each have five evenly spaced marks around their rims.
9. Grab two cones and put them next to each other so that they touch at marks. Stick the rims together at the marks with tape.
10. Stick another four cones around the remaining marks of one of the cones. You’ll end up with one cone, surrounded by a ring of five evenly spaced cones.
11. Go around the ring, matching marks on cones that are next to each other, taping them together. As you do this, the shape you’re making won’t sit flat anymore. It will begin to look like a star.
12. Repeat steps 9–11 to make a second star shape.
13. Take the two star-shapes and put them together, back-to-back to make a big ball.
14. Work along the join, matching marks and sticking them together with tape. This can get tricky, so it’s okay if you miss a few joins if they’re too hard to reach.
15. Congratulations! Enjoy your new decoration.

What’s happening?

Stars are fun and easy enough to draw on a piece of paper, but they’re a lot harder to make in three dimensions. If you look in stores, you’ll find that lots of star shaped decorations are sort of flat – they have a front and back, and they don’t have points going out in all directions like the ornament you’ve made.

So how did you create a star that works so well in three dimensions? The secret to this decoration is mathematical. There’s a hidden shape inside your star waiting to be discovered.

Each of the cones in your star is connected to five others. If you replace each of the 12 cones with a five-sided pentagon, you get a dodecahedron. This is a regular solid, similar to a cube or a triangular pyramid. All of its faces are the same shape, and all the sides and angles on each face are also the same. That’s why your star shape seems to fit together so neatly.

There are only five regular solids. The cube, triangular pyramid and dodecahedron are three of them. There is also the octahedron, made by sticking two square-based pyramids together base-to-base. The final regular solid is the icosahedron, which has 20 triangular sides.

Real-life maths

A good place to find model platonic solids is a board game shop – they usually have dice with 4, 6, 8, 12 and 20 sides that are all platonic solids!

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