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Platonic Planet

Age 14 to 16
Challenge Level

Why do this problem?

This problem gives learners the opportunity to use their visualisation skills to investigate different paths over the surface of a solid. Tackling the challenge of finding the shortest path offers a chance to practise trigonometry, as well as encouraging justification that there is no better path than the one they find.

Possible approach

Start with a cube and investigate nets and paths.
How about an octahedron?
Next, ask everyone to imagine they were standing on the surface of a dodecahedron. Ask what they can see if they are standing on a face, on an edge, or on a vertex.

One way of introducing the idea of paths on the surface of the dodecahedron is to have a model of a dodecahedron which can be drawn on and then unfolded into its net. Alternatively, the path could be drawn on a net and learners will need to imagine how the parts of the path will meet up when the net is folded up, to make sure it really is a closed path.

The pictures of nets shown in the problem are not necessarily the easiest nets to use to draw the path, so it is worth drawing out discussion of what would be a good net to use in order to aid visualisation.

Once everyone is happy with the idea of closed paths on the surface of the dodecahedron, learners can start to create their own paths and calculate the lengths. At each stage, they should be challenging themselves to see whether there is any way they can make the path shorter while still being able to see every part of the planet's surface.

In order to compare answers within the classroom, it is important for the class to decide how long one side of the planet will be. This could be a good opportunity to discuss the idea of working things out in terms of a unit length.

Key questions

How much of the planet's surface can I see if I stand at a face, an edge, a vertex?
Is it easier to investigate this problem using some nets rather than others?

Possible extension

The Dodecahedron provides an opportunity for further exploration of paths on the surface.

Possible support

Use accurate scale drawing to calculate path lengths rather than trigonometry.