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Fold the paper in half both ways to find the centre O.
Fold along the red line so A touches O.
Fold C to O similarly.
Fold B and D to O.
Next fold along PQ.
As the two halves come together,tuck the flap from corner D behind the flap from corner B to make 'pockets' (see the diagrams below).

Fold R and S up to the centre line EO, so that they meet to form a straight line and make a pentagon.
If you make 12 pentagons in this way and assemble them, using your 'flaps' and 'pockets', you can make a dodecahedron.
A4 paper has sides in the ratio$\sqrt2$ to $1$.
If you use A4 paper for this construction and try to make regular pentagons there is a small error in the angle at E. Find this error and find the dimensions of the paper which you would need to use to get an accurate regular pentagon and hence an accurate regular dodecahedron.
Extra Resources:
1) You can construct other platonic solids using paper and this article explains how.
2) Have a look at the October 2000 Article titled Classifying Solids using Angle Deficiency
3) You can download a demo version of Stella , a computer program which lets you create and view polyhedra on the screen, then print out the nets required to build your own models out of paper. Small Stella and Great Stella are available from the Stella Website.
4) Alternatively, print out the models from this pdf available at the British Crystallographic Association's Website
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?