The rhombic dodecahedron can be described as the convex hull of of the vertices $(\pm1,\pm1,\pm1)$ and permutations of $(\pm2,0,0)$ where all the preceeding $\pm$ symbols are independent. Wikipedia also informs us that the 4-dimensional analogue of the rhombic dodecahedron is the 24-cell.
What is the name of the $n$-dimensional polytope which is the convex hull of the vertices $(\pm1,\pm1,\ldots,\pm1)$ and permutations of $(\pm2,0,0,\ldots,0)$?
the Voronoi cell of the $D_n$ lattice. The elements of the lattice are those integer points with the sum of the integers even. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html#Dn
The lattice elements closest to the origin have $n-2$ entries zero, then two elements $\pm 1.$ The number of such short vectors, therefore the number of sides of the solid, is four times n choose 2, or $2n(n-1).$ In the Index, this is called the Kissing Number.