Michael Kleber wrote: << Russ Cox wrote:
complete the analogy:
square : rhombus :: cube : ____ ?
Do you mean to require that the twelve edges all have the same length, or that the six faces all have the same area? Or both? Or perhaps you want all faces to be congruent? Now I'm not even sure which Gareth's "equilateral parallelepiped" would best fit -- either the first or the last, I think.
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 <=> all of whose faces are congruent rhombi. The word for this 3D object is "rhombohedron". (It generalizes neatly to n dimensions. Not sure if there is a word for the n-dimensional object, but perhaps it ought to be "rhombotope".) All such rhombotopes can be obtained from a cube by applying an arbitrary dilatation to one main diagonal while keeping its orthogonal directions fixed, I ween. --Dan