# Thread: Projections from 3d to 4d

1. ## Projections from 3d to 4d

Hi there,

I'm doing a project where the goal is to map 4d polytopes to 3d in order to represent them graphically. In the project description it advises us to:

Choose a unit vector n in 4-dimensional space. Then the linear map x -> x - (x.n)n projects all of 4-dimensional space orthogonally on to the 3-dimensional subspace orthogonal to n. This does not solve the problem of producing a graphic of the projected object because we need to choose an orthonormal basis in that subspace to identify it with our usual coordinated is 3-dimensional space.

A way of doing this is to use the Gram-Schmidt process. Start with the basis n,e1,e2,e3 of 4-dim space, where e1,e2,e3 are the usual standard basis vectors. Apply G-S to get an orthonormal basis n,f1,f2,f3. Then v -> [ v.f1, v.f2, v.f3 ] gives the projection in coordinates.

There is one difficulty here that you will need to watch out for: if the last entry of n is zero then the original "basis" is not a basis. So one must instead in such cases make a different choice of three of the four possible standard basis vectors.

I think I understand the methodology behind this (though there is always the chance I don't) but a friend of mine doing the same thing has used v -> [ v.n, v.f2, v.f3 ] to map the projection co-ordinates (where v is a vertex of the 4d polytope); is this a viable thing to do just giving a different perspective of the projection, thus a different graphic representation or will in result in a poor representation of the 4d shape? If anyone could help clear this up for me I would be very grateful!

Thanks

2. There is nothing wrong with your friend's method. The aim of the suggested construction was to give the 3-D image of the polytope as viewed from the direction of n, whereas his method will give you the image as viewed from the direction of f1.

3. Oh yeah that makes sense, thank you!!