# Parallelopipedon

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

#### Comments

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 $\mathbf R^3$.

#### References

[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 |

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Parallelopipedon.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Parallelopipedon&oldid=31507