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A mathematical bauble

By David, 19 December 2016 Activity

Feeling festive? This cool mathematical bauble is fun to make, and looks great too!

A round ball made of colured paper.

Make this mathematical decoration and celebrate the end of the year!

You will need

  • Sheets of cardstock
  • Pencil
  • Cup
  • Scissors
  • Ruler
  • Glue

What to do

  1. First, cut out 20 paper circles. Start by turning your paper over, nice side down.
  2. Put the cup on the paper and trace it to make a circle. Move the cup and trace another circle, until you have 20 circles.
  3. Cut out the circles with a pair of scissors.

    Someone is using scissors to cut a circle out of paper.

    Cut out 20 paper circles.

  4. Take one circle, and surround it tightly with six more circles. Mark the point where every second circle touches the centre circle to get three evenly spaced marks.
  5. Draw straight lines between the points to make an equilateral triangle.

    Six circles surround a central circle. A triangle is drawn on the centre circle.

    Mark three equidistant points and join them with straight lines.

  6. Repeat steps 4 and 5 with all 20 circles.
  7. Turn the circles over so the nice side is facing up.
  8. Fold along each line to turn each circle into a triangle that has a rounded tab on each side.
  9. Time to start building! Take two triangles and put them side by side. Put some glue on the back side of one of the rounded tabs, and stick it to the back side of a rounded tab on another triangle to hold them together.

    Somoen is holding two pieces of paper together.

    Glue the triangles together using the tabs.

  10. Take three more triangles and add them to the first two so they all meet at one point, in a ‘c’ shape. Then attach the two triangles at the ends of the ‘c’ together. The shape will no longer lie flat – it should poke up in the middle. If it goes in at the middle, quickly unstick it and try again.

    A round, 3d shape made of five triangles.

    Stick five triangles together, and make sure it goes up in the middle.

  11. Repeat steps 9 and 10 to make a second group of five triangles.
  12. Take five more triangles and arrange them in a horizontal line. Make sure they all point forwards.
  13. In between the previous five triangles, put the remaining five triangles, pointing backwards. You’ll end up with a straight line of 10 triangles. Stick these triangles together.

    10 pieces of paper joined together in a line.

    Make a straight line of 10 triangles.

  14. When everything is dry, you can begin final assembly. First, take the straight line and bend it around into a ring. Then glue the two ends together.
  15. Put one of the groups of five triangles on top, so it matches the tabs on the ring. Glue it down and wait for the glue to dry.
  16. Finally, turn the ring over and stick the final group of five triangles on the other side of the ring. When the glue is dry, you’ll have a cool decoration!

What’s happening?

How do you make something round out of something flat? The secret is in the corners.

If you take a point on a flat piece of paper and divide it up into angles, they will always add together to make 360 degrees. If you look at the angles on this bauble, that’s not always the case.

At each vertex (3D corner), five equilateral triangles meet. Each of these is a 60 degree angle, so they add up to 300 degrees. This is why the triangles can’t lie flat – there’s not enough angle at this point.

This shape is an icosahedron, one of the 3D shapes known as platonic solids. They are all made up of regular flat shapes, and the angles at each vertex always add up to less than 360 degrees.

The five platonic solids are the triangular pyramid (three triangles meet at each vertex), the cube (three squares meet), the octahedron (four triangles meet), the dodecahedron (three pentagons meet) and the icosahedron (five triangles meet).

To make equilateral triangles lie flat, you need six of them to meet at each point. Then the angles add up to 360 degrees, and you’ll have what is known as a tessellation.

If you try putting seven triangles together, you’ll start making a completely different shape, known as a hyperbolic surface. They are pretty wacky – if you have a lot of time, try making one!

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