From Encyclopedia of Mathematics
Jump to: navigation, search

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 $3$-dimensional analogues of the Penrose tiles: They generate aperiodic tilings of $\mathbf R^3$.


[a1] B. Grünbaum, "Convex polytopes" , Wiley (1967)
[a2] P. Kramer, R. Neri, "On periodic and non-periodic space fillings of $E^m$ 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