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Octahedron question

I'm trying to get some geometry practice in as I'm not very good at it, in particular 3d geometry.

Show that the angle between any two faces of a regular ocrahedron is arccos(-1/3). (Regular octahedron is a polyhedron with 8 eight faces each an equalaterial triangle).

Find the ratio of the volume of a regular octahedron to the volume of the cube whose vertices are the centers of the faces of the octahedron.

The first bit doesn't seem too bad, but my main issue is coming up with a sketch. Do you generally sketch in 2D? also how do you sketch it? Sorry for all the questions, I just find 3d geometry so confusing.

Reply 1

maths learner

Here is a rough image of what I've constructed for the Octahedron, but now I'm a bit unsure on how to proceed. Obviously the angles of each triangle are 60 degrees as they are all equalaterial...

Bump.

Bump!

Original post by maths learner

Bump!

Problems like this can often be easier if you work in coordinates. Let ABCDEF be an octahedron. Since the cube and octahedron are dual, in the sense that one can be constructed using the centre of each face of the other, we may first consider a cube of side length 2 with one vertex at the origin.

This allows us to see (literally :P) that we may take the position vectors of A,B,C,D to be (1,1,0),(1,0,1), (0,1,1), (1,2,1), (2,1,1), (1,1,2). Then if you know about the scalar and vector products, you can get the angle between the faces easily.

Of course, this method doesn't help with visualisation very much, but it does allow you to solve the question. At university, 'geometry' can get very abstract and visualisation can become hard. Therefore we have additional techniques (such as abstract algebra) to aid us in doing calculations. Despite this, however, visualisation is obviously very important in geometry, and so I would recommend spending some time practising it.