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Parallelotope


Move a point Pi_0 along a line from 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 dimensions is then called a parallelotope. Pi_n has 2^n vertices and

 N_k=2^(n-k)(n; k)

Pi_ks, where (n; 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

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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.

Referenced on Wolfram|Alpha

Parallelotope

Cite this as:

Weisstein, Eric W. "Parallelotope." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Parallelotope.html

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