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This shape is a lie!

By David, 30 May 2019 Activity

A paper model of a chamfered cubeThis shape has a little secret. It’s known as a chamfered cube, and it’s not as perfect as it first seems…

You will need

What to do

  1. Carefully cut along the outline of the shape on the printout. You want it all to be one piece.
  2. Fold and unfold along each of the black lines.
  3. Before you start sticking, see if you can roughly curl the shape so it appears ball-like. This will be an indicator of what your final shape will look like.
  4. Using small pieces of sticky tape, start sticking the panels together. If you put the tape on the inside of the shape you’re building, it will look better but it will also be a bit trickier to build!
  5. Here are a few guidelines to make sure everything is going right:
    • Every square will be surrounded by four hexagons.
    • Every hexagon will touch two squares, and the squares will be on opposite sides of the hexagon.
    • At each corner, there will be either three hexagons or two hexagons and a square.
  6. If you’re taping on the inside of the shape, the last bit can be quite difficult. Try folding one hexagon the wrong way, then stick bits of sticky tape where that hexagon needs to go. Then fold the paper back into place onto the sticky tape you prepared.

What’s happening?

The common name for the object you’ve made is a chamfered cube. If you take a cube and then slice off all the edges, this is the shape you’ll get. So, what makes it special?

This three dimensional object looks like it’s made from squares and regular hexagons. But if you look carefully, you’ll notice that the hexagons are a bit squished. The hexagon sides are all the same length, but the angles are not all the same. If you try to make this shape with perfectly regular squares and hexagons, the angles just don’t work, and it won’t make a nice ball-like shape.

Three dimensional objects made from regular shapes are called Johnson solids. This shape is a near-miss Johnson solid. It’s close to being one, but not quite there.

Digging deeper

This not-quite-right shape is a not a sign that something is wrong. Maths is full of strange things that don’t quite work the way you might expect!

There is a solid shape you can make with regular squares and hexagons. It’s called a truncated octahedron and it looks a bit like this shape. The main difference is that each hexagon touches three squares rather than two.

So if this shape made you feel a bit weird, that’s okay – there is a more perfect shape for you out there.

More information:

Fold a similar shape, used as a dice by the Ancient Chinese

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