Difference between revisions of "Parallelopipedon"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
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. | ||
| Line 8: | Line 9: | ||
====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 | + | 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 | + | <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=31507
Parallelopipedon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallelopipedon&oldid=31507
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article