Parallelotope

Move a point $\Pi_0$ along a Line for an initial point to a final point. It traces out a Line Segment $\Pi_1$. When $\Pi_1$ is translated from an initial position to a final position, it traces out a Parallelogram $\Pi_2$. When $\Pi_2$ is translated, it traces out a Parallelepiped $\Pi_3$. The generalization of $\Pi_n$ to $n$-D is then called a parallelotope. $\Pi_n$ has $2^n$ vertices and

$$N_k=2^{n-k}{n\choose k}$$

$\Pi_k$s, where ${n\choose k}$ is a Binomial Coefficient and $k=0$, 1, ..., $n$ (Coxeter 1973). These are also the coefficients of $(x+2)^n$.

See also Honeycomb, Hypercube, Orthotope, Parallelohedron

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 122-123, 1973.

Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991.

Zaks, J. ``Neighborly Families of Congruent Convex Polytopes.'' Amer. Math. Monthly 94, 151-155, 1987.