A Hyperrectangle or hypercube is in the geometry of the generalization of the rectangle and the cuboid to any number of dimensions. The hypercube is a special case of this.
definition An axially parallel hyper rectangle in -dimensional space is the Cartesian product of real intervals with for , that is
R.
{\ displaystyle R}
n
{\ displaystyle n}
R.
n
{\ displaystyle \ mathbb {R} ^ {n}}
n
{\ displaystyle n}
[
a
i
,
b
i
]
{\ displaystyle [a_ {i}, b_ {i}]}
a
i
≤
b
i
{\ displaystyle a_ {i} \ leq b_ {i}}
i
=
1
,
...
,
n
{\ displaystyle i = 1, \ dotsc, n}
R.
=
∏
i
=
1
n
[
a
i
,
b
i
]
=
[
a
1
,
b
1
]
×
[
a
2
,
b
2
]
×
⋯
×
[
a
n
,
b
n
]
{\ displaystyle R = \ prod _ {i = 1} ^ {n} [a_ {i}, b_ {i}] = [a_ {1}, b_ {1}] \ times [a_ {2}, b_ { 2}] \ times \ dotsb \ times [a_ {n}, b_ {n}]}
In general, a hyper rectangle is a figure that is congruent with an axially parallel hyper rectangle.
Examples For one obtains an interval of a rectangle and a square .
n
=
1
{\ displaystyle n = 1}
n
=
2
{\ displaystyle n = 2}
n
=
3
{\ displaystyle n = 3}
For the special case that all intervals equal to the unit interval are, you get the unit hypercube
[
0
,
1
]
{\ displaystyle [0,1]}
R.
=
∏
i
=
1
n
[
0
,
1
]
=
[
0
,
1
]
n
{\ displaystyle R = \ prod _ {i = 1} ^ {n} [0,1] = [0,1] ^ {n}}
properties Limiting elements Every -dimensional hyper-rectangle with has
n
{\ displaystyle n}
n
≥
2
{\ displaystyle n \ geq 2}
2
n
{\ displaystyle 2 ^ {n}}
n
2
n
-
1
{\ displaystyle n2 ^ {n-1}}
at right angles , and
2
n
{\ displaystyle 2n}
n
-
1
{\ displaystyle n-1}
In general, a -dimensional hyper-rectangle of
n
{\ displaystyle n}
(
n
k
)
⋅
2
n
-
k
{\ displaystyle {\ binom {n} {k}} \ cdot 2 ^ {nk}}
Hyper-rectangles of dimension bounded, where is.
k
{\ displaystyle k}
k
∈
{
0
,
...
,
n
-
1
}
{\ displaystyle k \ in \ {0, \ dotsc, n-1 \}}
Volume and surface
The volume of a hyper rectangle is
R.
{\ displaystyle R}
vol
(
R.
)
=
∏
i
=
1
n
(
b
i
-
a
i
)
=
(
b
1
-
a
1
)
⋅
(
b
2
-
a
2
)
⋯
(
b
n
-
a
n
)
{\ displaystyle \ operatorname {vol} (R) = \ prod _ {i = 1} ^ {n} (b_ {i} -a_ {i}) = (b_ {1} -a_ {1}) \ cdot ( b_ {2} -a_ {2}) \ dotsm (b_ {n} -a_ {n})}
This is the starting point for determining the volumes much more general levels, as in the construction of the dimensional Lebesgue measure in the measure theory is clear. The surface area is
n
{\ displaystyle n}
vol
(
∂
R.
)
=
2
∑
j
=
1
n
∏
i
=
1
i
≠
j
n
(
b
i
-
a
i
)
{\ displaystyle \ operatorname {vol} (\ partial R) = 2 \ sum _ {j = 1} ^ {n} \ prod _ {i = 1 \ atop i \ neq j} ^ {n} (b_ {i} -a_ {i})}
See also Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![[a_ {i}, b_ {i}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1adde530cc8cd17a451c905e20cfe882e73fdd)
![{\ displaystyle R = \ prod _ {i = 1} ^ {n} [a_ {i}, b_ {i}] = [a_ {1}, b_ {1}] \ times [a_ {2}, b_ { 2}] \ times \ dotsb \ times [a_ {n}, b_ {n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1184da3bc4afdb4cf85758e191ce579a20f983ac)
![[0.1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![R = \ prod _ {{i = 1}} ^ {{n}} [0,1] = [0,1] ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790aade53d3bc51076bc5d493a9bc263c9561e49)
