Sorry to bring the discussion back to geometry instead of news group interfaces. Bug Dan Asimov wrote:
Technically there are several ways to generalize the rhombus to 3D.
But the most natural and symmetric way seems to me for it to be a parallepiped
all of whose edges are the same length
<=>
all of whose faces are rhombi
I agree with you so far, of course, but:
<=>
all of whose faces are congruent rhombi.
I don't think that's equivalent at all. Given any three vectors a,b,c of unit length, the parallepiped whose eight vertices are +-a+-b+-c has all faces rhombi of edge length 2, but it generally has three different rhombic face shapes.
The word for this 3D object is "rhombohedron".
I agree that "rhombohedron" refers to one of these two things, but I'm not sure which, and in fact I'm not even convinced that different people all agree on which it is. --Michael Kleber ps. Evidence for this last hypothesis: the top two google hits for definitions of "rhombohedron" are MathWorld, "A parallelepiped bounded by six congruent rhombi," and answers.com quoting the American Heritage Dictionary, "A prism with six faces, each a rhombus." -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.