View source for Parallelopipedon

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A [[Hexahedron|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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071450a.gif" />

Figure: p071450a



====Comments====
A parallelopipedon is a special case of a [[Parallelohedron|parallelohedron]] and of a [[Parallelotope|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 $\mathbf R^3$.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Grünbaum,   "Convex polytopes" , Wiley  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Kramer,   R. Neri,   "On periodic and non-periodic space fillings of $E^m$ obtained by projection"  ''Acta Cryst.'' , '''A40'''  (1984)  pp. 580–587</TD></TR></table>