Single cube

A unit cube is a cube whose edge is a unit segment , respectively, a face is a unit square . In a rectangular coordinate system , it is usually assumed that one vertex is at the origin , all edges are parallel to the coordinate axes, and the entire cube is in the first octant , that is, so that the coordinates of the vertices are:

Single cube

(0;0;0),(1;0;0),(1;1;0),(1;1;1),(0;1;1),(0;1;0),(0;0;1),(1;0;1)

{\ displaystyle (0; 0; 0), (1; 0; 0), (1; 1; 0), (1; 1; 1), (0; 1; 1), (0; 1; 0) , (0; 0; 1), (1; 0; 1)}

{\ displaystyle (0; 0; 0), (1; 0; 0), (1; 1; 0), (1; 1; 1), (0; 1; 1), (0; 1; 0) , (0; 0; 1), (1; 0; 1)}

.

The volume of a single cube is 1, the surface area is 6, the length of the longest diagonal is ${\sqrt {3}}$ ${\ displaystyle {\ sqrt {3}}}$ ${\ sqrt {3}}$ .

Single hypercube ( single $n$ ${\ displaystyle n}$ $n$ cube ) - $n$ ${\ displaystyle n}$ $n$ -dimensional generalization of a unit cube, a hypercube with edges of length 1, and (if mentioned in the context of a rectangular coordinate system) lying $n$ ${\ displaystyle n}$ $n$ edges on the coordinate axes, one of the vertices located at the origin and located in the first orthant . Hypervolume $n$ ${\ displaystyle n}$ $n$ -dimensional hypercube - 1, surface hyper area - $2\cdot n$ ${\ displaystyle 2 \ cdot n}$ ${\ displaystyle 2 \ cdot n}$ , the longest diagonal has a length ${\sqrt {n}}$ ${\ displaystyle {\ sqrt {n}}}$ ${\ sqrt n}$ .

Identify unit $n$ ${\ displaystyle n}$ $n$ -cube can be a Cartesian product of unit segments:

[0,1]^{n}=[0,1]\times [0,1]\times [0,1]\times \dots \times [0,1]

{\ displaystyle [0,1] ^ {n} = [0,1] \ times [0,1] \ times [0,1] \ times \ dots \ times [0,1]}

{\ displaystyle [0,1] ^ {n} = [0,1] \ times [0,1] \ times [0,1] \ times \ dots \ times [0,1]}

.

Infinite-dimensional generalizations of a unit hypercube - a Hilbert brick , defined as the product of a countable number of unit segments, and an even more general Tikhonov cube , which is a product of unit segments indexed by an arbitrary (possibly uncountable) set.

Literature

R. Engelking. General topology. - M .: Mir , 1986 .-- S. 130 .-- 752 p.

Single cube

Literature

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