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Difference between revisions of "Parallelotope"

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$$  
 
$$  
\mathbf h  =  \sum _ { i= } 1 ^ { p }  x  ^ {i} \mathbf a _ {i} ,
+
\mathbf h  =  \sum _ { i=1 } ^ { p }  x  ^ {i} \mathbf a _ {i} ,
 
$$
 
$$
  
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are fixed vectors in an [[Affine space|affine space]]  $  A $
 
are fixed vectors in an [[Affine space|affine space]]  $  A $
 
of dimension  $  n $.  
 
of dimension  $  n $.  
They are called the generators of the parallelotope and coincide with some of the edges of the parallelotope. All remaining edges of the parallelotope are parallel to them. If the generators of the parallelotope are linearly independent (dependent), then the parallelotope is called  $  p $-
+
They are called the generators of the parallelotope and coincide with some of the edges of the parallelotope. All remaining edges of the parallelotope are parallel to them. If the generators of the parallelotope are linearly independent (dependent), then the parallelotope is called  $  p $-dimensional, or non-degenerate (degenerate). Degenerate parallelotopes are parallel projections of some  $  p $-dimensional parallelotope onto a plane of dimension  $  k \leq  p- 1 $.  
dimensional, or non-degenerate (degenerate). Degenerate parallelotopes are parallel projections of some  $  p $-
+
A non-degenerate parallelotope determines a supporting  $  p $-dimensional plane. Such a parallelotope for  $  p= 2 $
dimensional parallelotope onto a plane of dimension  $  k \leq  p- 1 $.  
 
A non-degenerate parallelotope determines a supporting  $  p $-
 
dimensional plane. Such a parallelotope for  $  p= 2 $
 
 
is a [[Parallelogram|parallelogram]], and for  $  p= 3 $
 
is a [[Parallelogram|parallelogram]], and for  $  p= 3 $
 
is a [[Parallelopipedon|parallelopipedon]].
 
is a [[Parallelopipedon|parallelopipedon]].
  
Two non-degenerate parallelotopes are said to be parallel if their supporting planes are parallel. For parallel parallelotopes it is possible to compare their  $  p $-
+
Two non-degenerate parallelotopes are said to be parallel if their supporting planes are parallel. For parallel parallelotopes it is possible to compare their  $  p $-dimensional  "volume"  (although there need not be a metric in  $  A $).  
dimensional  "volume"  (although there need not be a metric in  $  A $).  
+
For the numerical measure of the ratio of the  $  p $-dimensional  "volume"  of the parallelotope with generators  $  \mathbf a _ {1} \dots \mathbf a _ {p} $
For the numerical measure of the ratio of the  $  p $-
+
to the  $  p $-dimensional  "volume"  of the parallel parallelotope with generators  $  \mathbf b _ {1} \dots \mathbf b _ {p} $,  
dimensional  "volume"  of the parallelotope with generators  $  \mathbf a _ {1} \dots \mathbf a _ {p} $
 
to the  $  p $-
 
dimensional  "volume"  of the parallel parallelotope with generators  $  \mathbf b _ {1} \dots \mathbf b _ {p} $,  
 
 
the scalar  $  \mathop{\rm det} ( x _ {j}  ^ {i} ) $
 
the scalar  $  \mathop{\rm det} ( x _ {j}  ^ {i} ) $
 
is used, where  $  ( x _ {j}  ^ {i} ) $
 
is used, where  $  ( x _ {j}  ^ {i} ) $
is the  $  ( p \times p) $-
+
is the  $  ( p \times p) $-matrix which transforms  $  ( \mathbf b _ {1} \dots \mathbf b _ {p} ) $
matrix which transforms  $  ( \mathbf b _ {1} \dots \mathbf b _ {p} ) $
 
 
to  $  ( \mathbf a _ {1} \dots \mathbf a _ {p} ) $,  
 
to  $  ( \mathbf a _ {1} \dots \mathbf a _ {p} ) $,  
 
i.e.
 
i.e.
  
 
$$  
 
$$  
\mathbf a _ {j}  =  \sum _ { i= } 1 ^ { p }  x _ {j}  ^ {i} \mathbf b _ {i} ,\ \  
+
\mathbf a _ {j}  =  \sum _ { i=1 } ^ { p }  x _ {j}  ^ {i} \mathbf b _ {i} ,\ \  
 
1 \leq  j \leq  p.
 
1 \leq  j \leq  p.
 
$$
 
$$
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then the square of the  $  p $-
 
then the square of the  $  p $-
 
dimensional volume of the parallelotope with generators  $  \mathbf a _ {1} \dots \mathbf a _ {p} $
 
dimensional volume of the parallelotope with generators  $  \mathbf a _ {1} \dots \mathbf a _ {p} $
is equal to the [[Determinant|determinant]] of the  $  ( p \times p) $-
+
is equal to the [[Determinant|determinant]] of the  $  ( p \times p) $-dimensional [[Gram matrix|Gram matrix]] with entries  $  ( \mathbf a _ {i} , \mathbf a _ {j} ) $
dimensional [[Gram matrix|Gram matrix]] with entries  $  ( \mathbf a _ {i} , \mathbf a _ {j} ) $(
+
(cf. also [[Gram determinant|Gram determinant]]).
cf. also [[Gram determinant|Gram determinant]]).
 
  
 
The concept of a parallelotope is closely connected with the concept of a [[Poly-vector|poly-vector]].
 
The concept of a parallelotope is closely connected with the concept of a [[Poly-vector|poly-vector]].

Latest revision as of 15:25, 13 January 2021


A set of points whose radius vectors have the form

$$ \mathbf h = \sum _ { i=1 } ^ { p } x ^ {i} \mathbf a _ {i} , $$

with the possible values $ 0 \leq x ^ {i} \leq 1 $, $ 1 \leq i \leq p $. Here $ \mathbf a _ {1} \dots \mathbf a _ {p} $ are fixed vectors in an affine space $ A $ of dimension $ n $. They are called the generators of the parallelotope and coincide with some of the edges of the parallelotope. All remaining edges of the parallelotope are parallel to them. If the generators of the parallelotope are linearly independent (dependent), then the parallelotope is called $ p $-dimensional, or non-degenerate (degenerate). Degenerate parallelotopes are parallel projections of some $ p $-dimensional parallelotope onto a plane of dimension $ k \leq p- 1 $. A non-degenerate parallelotope determines a supporting $ p $-dimensional plane. Such a parallelotope for $ p= 2 $ is a parallelogram, and for $ p= 3 $ is a parallelopipedon.

Two non-degenerate parallelotopes are said to be parallel if their supporting planes are parallel. For parallel parallelotopes it is possible to compare their $ p $-dimensional "volume" (although there need not be a metric in $ A $). For the numerical measure of the ratio of the $ p $-dimensional "volume" of the parallelotope with generators $ \mathbf a _ {1} \dots \mathbf a _ {p} $ to the $ p $-dimensional "volume" of the parallel parallelotope with generators $ \mathbf b _ {1} \dots \mathbf b _ {p} $, the scalar $ \mathop{\rm det} ( x _ {j} ^ {i} ) $ is used, where $ ( x _ {j} ^ {i} ) $ is the $ ( p \times p) $-matrix which transforms $ ( \mathbf b _ {1} \dots \mathbf b _ {p} ) $ to $ ( \mathbf a _ {1} \dots \mathbf a _ {p} ) $, i.e.

$$ \mathbf a _ {j} = \sum _ { i=1 } ^ { p } x _ {j} ^ {i} \mathbf b _ {i} ,\ \ 1 \leq j \leq p. $$

If an inner product is defined in $ A $, then the square of the $ p $- dimensional volume of the parallelotope with generators $ \mathbf a _ {1} \dots \mathbf a _ {p} $ is equal to the determinant of the $ ( p \times p) $-dimensional Gram matrix with entries $ ( \mathbf a _ {i} , \mathbf a _ {j} ) $ (cf. also Gram determinant).

The concept of a parallelotope is closely connected with the concept of a poly-vector.

References

[1] P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian)
[2] D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian)
[3] C. Pisot, M. Zamansky, "Mathématiques générales: algèbre-analyse" , Dunod (1966)

Comments

Parallelotopes are special types of zonotopes (cf. Zonohedron). They play a basic role in the geometry of numbers and in the theory of lattice covering and packing.

References

[a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2] B. Grünbaum, "Convex polytopes" , Wiley (1967)
How to Cite This Entry:
Parallelotope. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallelotope&oldid=48119
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article