STEP I/2007/5: the regular octahedron

Above is half a rectangular octahedron being constructed in the car park of the industrial estate where I work. It’s being constructed as an artwork, In the Eye of the Storm, by Tom Paine and the Structured Light group, and includes LED lights placed along all the lengths of wood which change colour and switch on and off in a way devised by Tom. It’ll be on show in five art festivals this year.

And here is the STEP question. When it was in the STEP exam in 2007, and when we offered it to the students on the 2018-9 Greenwich STEP course as a practice problem, few students got anywhere with it. Mya Titus, however, cracked the problem without any help.


STEP I/2007/5:

(i) Show that the angle between any two faces of a regular octahedron is arccos(−13)

(ii) Find the ratio of the volume of a regular octahedron to the volume of a cube whose vertices are the centres of the faces of the octahedron.

This diagram of a half-octahedron helps with a solution. It’s easier working with it than with a diagram of a full octahedron.

Click here for a solution.

Here’s a full octahedron.