# Puzzle Page

By Faux, Geoff | Mathematics Teaching, January 2008 | Go to article overview

# Puzzle Page

Faux, Geoff, Mathematics Teaching

If you glue a tetrahedron onto f each of the eight faces of an octahedron, you get a stellated octahedron, called a stellaoctangula. Here are some visualisation problems based on a stella-octangula, and it might be helpful to make one before embarking on them.

The idea is to work with pictures in your mind. These problems are ideal to do in bed when sleep does not come and there is no paper to hand. Setting them as visualisation problems suggests that the answers may be simple fractions.

1 The stella-octangula can be seen as two interlocking tetrahedra. Find the ratio of the volume of one of the tetrahedra to the central octahedron.

2 The model shows that an octahedron will fit inside a cube with its six vertices at the midpoints of the six faces of the cube. Find the ratio of the volume of the octahedron to the volume of the containing cube. And inside the octahedron there will be another cube ... so can we imagine how much smaller a face of this cube is than the face of the outside cube? And the next one? etc

3 A cubeoctahedron (see MT205, pages 24-25) can be seen as a cube with a number of tetrahedra removed. What is the ratio of the volume of the cubeoctahedron to the containing cube?

4 A cubeoctahedron can equally be seen as an octahedron with a number of square-based pyramids removed. What is the ratio of the volume of the cubeoctahedron to the containing octahedron. …

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