In 3D, start with 3 distinct vectors and 3 scalars. For every matching of the scalars to vectors (i.e., every permutation), compute the weighted sum (linear combination). One gets 1, 3, or 6 new vectors, depending on how many of the scalars were distinct.
Given a polyhedron whose faces are triangles, this can be done on each face, using the vertices of each face as the input vectors. With the regular octahedron, the input vectors are the positive and negative coordinate axes.
Not sure what this might be useful for. If the scalars sum to unity, the new vectors lie on the face. Or we could rescale the vector endpoints to a sphere. Or compute the convex hull, though the result might not be polyhedral when in three dimensions or more.
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