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Difference between revisions of "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.
 
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.
  
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====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071450/p0714501.png" />-dimensional analogues of the Penrose tiles: They generate aperiodic tilings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071450/p0714502.png" />.
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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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071450/p0714503.png" /> obtained by projection"  ''Acta Cryst.'' , '''A40'''  (1984)  pp. 580–587</TD></TR></table>
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<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>

Latest revision as of 16:21, 11 April 2014

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
How to Cite This Entry:
Parallelopipedon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallelopipedon&oldid=12184
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article