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A hexahedron whose opposite faces are pairwise parallel. A parallelopipedon has 8 vertices and 12 edges; its faces are pairwise congruent parallelograms. A parallelopipedon is called rectangular if the 6 faces are rectangles; a parallelopipedon all faces of which are squares is called a cube. The volume of a parallelopipedon is equal to the product of the area of its base and its height.

Figure: p071450a


A parallelopipedon is a special case of a parallelohedron and of a parallelotope. Two special parallelopipeda, namely the golden rhombohedra or Ammann rhombohedra, play an essential role in the theory of quasi-crystals, because they are the -dimensional analogues of the Penrose tiles: They generate aperiodic tilings of .


[a1] B. Grünbaum, "Convex polytopes" , Wiley (1967)
[a2] P. Kramer, R. Neri, "On periodic and non-periodic space fillings of obtained by projection" Acta Cryst. , A40 (1984) pp. 580–587
How to Cite This Entry:
Parallelopipedon. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article