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Куб
Avengers Tesseract2012

A cube (also known as a hexahedron) is the 3-dimensional hypercube. It is also the only platonic solid that can perfectly tessellate space by itself in a honeycomb, forming the cubic honeycomb. Under the elemental naming scheme it is called a geohedron. Among the Platonic solids the cube represents Earth since it is solid and unchanging.

It has the Schläfli symbol

$ \{4,3\} $ , meaning that it is made of squares, three of which meet at each vertex. It can also be represented by the Schläfli symbols

$ { \{ \} }^{ 3 } $ as it is the product of three line segments,

$ \{ 4\} \times \{ \} $ as it is the product of a square and a line segment (in other words, a square-based prism) and

$ t\{ 2,4\} $ as it is a truncated square hosohedron.

Its Bowers acronym is also "cube".

Hypercube Product

A cube can be expressed as the product of hypercubes in 3 different ways:

$ \{4,3\} $ - cube

As a regular cube, the subfacets and hypervolumes depend only on a single parameter, the edge length l. This is the most symmetrical form of the cube, and is a uniform, regular platonic solid.

$ \{4\} \times \{\} $ - square prism

As a square prism, created by the Cartesian product of a regular square and a line segment, the cube has square prismatic symmetry (D4h) with the abstract group Dih4 × Z2. The hypervolumes of a square prism depend on two parameters: the edge length a of the square, and the height of the prism b.

The hypervolumes are:

  • edge length =$ 4 \left( 2a + b \right) $
  • surface area =$ 2a \left( a + 2b \right) $
  • surcell volume =$ a^2 b $(when a=b, this becomes the regular cube.)

$ \{\}^3 $ - line prism prism

As a line prism prism, also known as a rectangular cuboid, created by the Cartesian product of three seperate line segments, the cube has digonal prismatic symmetry. The hypervolumes of a rectangular cuboid depend on three parameters, the lengths a, b and c of the line segments.

The hypervolumes are:

  • edge length =$ 4 \left( a + b + c \right) $
  • surface area =$ 2 \left( ab + ac + bc \right) $
  • surcell volume =$ abc $ (when a=b, b=c xor a=c, this becomes the square prism. When a=b=c, this becomes the regular cube)

Symbols

A cube can be given various Dynkin symbols, including:

  • x4o3o (regular)
  • x x4o (square prism)
  • x x x(rectangular prism)
  • qo3oo3oq&#zx (2-coloring of vertices)
  • x2s4s, x2s8o (variations of the above)
  • s2s4x (two rectangles and 4 trapezoids)
  • xx4oo&#x (square frustum)
  • xx xx&#x (rectangle frustum)
  • oqoo3ooqo&#xt (triangular antitegum)
  • xxx oqo&#xt (prism of kite)
  • xx qo oq&#zx (rhombic prism)

Structure and Sections

The cube is composed of many squares stacked on each other, making it a prism with a square as the base. It is composed of two parallel squares linked by a ring of four squares. Three squares join at each corner.

When viewed from a square face, it appears as a constant sized square. When viewed from an edge, it looks like a line expanding to a rectangle and back. Finally, when viewed from a corner, it is a point that expands into an equilateral triangle, then truncates to various hexagons, then goes back to a triangle (oriented the other way) which then shrinks.

Hypervolumes

Subfacets

Only 4D creatures and above could see all of a cube.

Radii

  • Vertex radius:$ \frac{\sqrt{3}}{2}l $
  • Edge radius:$ \frac{\sqrt{2}}{2}l $
  • Face radius:$ 1/2l $

Angles

  • Dihedral angle: 90º

Vertex coordinates

The vertex coordinates of a cube of side length 2 are (±1,±1,±1).

Equations

All points on the surface of a cube of side 2 can be given by the equation

$ \max(x^2,y^2,z^2) = 1 $

Notations

  • Toratopic notation:$ ||| $
  • Tapertopic notation:$ 111 $

Related shapes

Coordinate System

The coordinate system corresponding to the cube is called realm cartesian coordinates, with the three coordinates being

$ \left(x, y, z\right) $ . The length elements of cartesian coordinates are simply

$ dx $ ,

$ dy $ and

$ dz $ , giving a line element of

$ ds = dx \hat{x} + dy \hat{y} + dz \hat{z} $ with a length

$ ds^2 = dx^2 + dy^2 + dz^2 $ . The surface elements, giving the changes in area for small changes in x, y and z, are

$ dx dy $ ,

$ dx dz $ and

$ dy dz $ . The volume element, giving the change in volume for small changes in x, y and z, is

$ dx dy dz $ .

See Also

Regular polyhedra
Convex regular polyhedra: tet · cube · oct · doe · ike

Self-intersecting regular polyhedra: gad · sissid · gike · gissid

Dimensionality Negative First Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
Simplex

$ \{3^{n-1}\} $

Null polytope

$ \emptyset $

Point

$ () $
$ \mathbb{B}^0 $

Line segment

$ \{\} $
$ \mathbb{B}^1 $

Triangle

$ \{3\} $

Tetrahedron

$ \{3^2\} $

Pentachoron

$ \{3^3\} $

Hexateron

$ \{3^4\} $

Heptapeton

$ \{3^5\} $

Octaexon

$ \{3^6\} $

Enneazetton

$ \{3^7\} $

Decayotton

$ \{3^8\} $

Hendecaxennon

$ \{3^9\} $

Dodecadakon

$ \{3^{10}\} $

Tridecahendon

$ \{3^{11}\} $

Tetradecadokon

$ \{3^{12}\} $

Pentadecatradakon

$ \{3^{13}\} $

Hexadecatedakon

$ \{3^{14}\} $

Heptadecapedakon

$ \{3^{15}\} $

... Omegasimplex

$ \{3^{\aleph_0}\} $

Cross

$ \{3^{n-2},4\} $

Square

$ \{4\} $

Octahedron

$ \{3, 4\} $

Hexadecachoron

$ \{3^2, 4\} $

Pentacross

$ \{3^3, 4\} $

Hexacross

$ \{3^4, 4\} $

Heptacross

$ \{3^5, 4\} $

Octacross

$ \{3^6, 4\} $

Enneacross

$ \{3^7, 4\} $

Dekacross

$ \{3^8, 4\} $

Hendekacross

$ \{3^9, 4\} $

Dodekacross

$ \{3^{10}, 4\} $

Tridekacross

$ \{3^{11}, 4\} $

Tetradekacross

$ \{3^{12}, 4\} $

Pentadekacross

$ \{3^{13}, 4\} $

Hexadekacross

$ \{3^{14}, 4\} $

... Omegacross

$ \{3^{\aleph_0}, 4\} $

Hydrotopes

$ \{3^{n-2}, 5\} $

Pentagon

$ \{5\} $

Icosahedron

$ \{3, 5\} $

Hexacosichoron

$ \{3^2, 5\} $

Order-5 pentachoric tetracomb

$ \{3^3, 5\} $

Hypercube

$ \{4, 3^{n-2}\} $

Square

$ \{4\} $

Cube

$ \{4, 3\} $

Tesseract

$ \{4, 3^2\} $

Penteract

$ \{4, 3^3\} $

Hexeract

$ \{4, 3^4\} $

Hepteract

$ \{4, 3^5\} $

Octeract

$ \{4, 3^6\} $

Enneract

$ \{4, 3^7\} $

Dekeract

$ \{4, 3^8\} $

Hendekeract

$ \{4, 3^9\} $

Dodekeract

$ \{4, 3^{10}\} $

Tridekeract

$ \{4, 3^{11}\} $

Tetradekeract

$ \{4, 3^{12}\} $

Pentadekeract

$ \{4, 3^{13}\} $

Hexadekeract

$ \{4, 3^{14}\} $

... Omegeract

$ \{4, 3^{\aleph_0}\} $

Cosmotopes

$ \{5, 3^{n-2}\} $

Pentagon

$ \{5\} $

Dodecahedron

$ \{5, 3\} $

Hecatonicosachoron

$ \{5, 3^2\} $

Order-3 hecatonicosachoric tetracomb

$ \{5, 3^3\} $

Hyperball

$ \mathbb B^n $

Disk

$ \mathbb B^2 $

Ball

$ \mathbb B^3 $

Gongol

$ \mathbb B^4 $

Pentorb

$ \mathbb B^5 $

Hexorb

$ \mathbb B^6 $

Heptorb

$ \mathbb B^7 $

Octorb

$ \mathbb B^8 $

Enneorb

$ \mathbb B^9 $

Dekorb

$ \mathbb B^{10} $

Hendekorb

$ \mathbb B^{11} $

Dodekorb

$ \mathbb B^{12} $

Tridekorb

$ \mathbb B^{13} $

Tetradekorb

$ \mathbb B^{14} $

Pentadekorb

$ \mathbb B^{15} $

Hexadekorb

$ \mathbb B^{16} $

... Omegaball

$ \mathbb B^{\aleph_0} $

$ \{4,2\} $ $ \{4,3\} $ $ \{4,4\} $ $ \{4,5\} $ $ \{4,6\} $ ... $ \{4,\aleph_0\} $
Square dihedron Cube Square tiling Order-5 square tiling Order-6 square tiling ... Infinite-order square tiling
$ \{2,3\} $ $ \{3,3\} $ $ \{4,3\} $ $ \{5,3\} $ $ \{6,3\} $ $ \{7,3\} $ $ \{8,3\} $ ... $ \{\aleph_0,3\} $
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling
$ {t}_{0,1} \{2,2\} $ $ {t}_{0,1} \{2,3\} $ $ {t}_{0,1} \{2,4\} $ $ {t}_{0,1} \{2,5\} $ $ {t}_{0,1} \{2,6\} $ $ {t}_{0,1} \{2,7\} $ $ {t}_{0,1} \{2,8\} $ ... $ {t}_{0,1} \{2,\aleph_0\} $
Digonal prism Triangular prism Cube Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism ... Apeirogonal prism
Regular
$ t_0 \{4,3\} $
Rectified
$ t_1 \{4,3\} $
Birectified
$ t_2 \{4,3\} $
Truncated
$ t_{0,1} \{4,3\} $
Bitruncated
$ t_{1,2} \{4,3\} $
Cantellated
$ t_{0,2} \{4,3\} $
Cantitruncated
$ t_{0,1,2} \{4,3\} $
Cube Cuboctahedron Octahedron Truncated cube Truncated octahedron Rhombicuboctahedron Great rhombicuboctahedron
Regular
$ t_0 \{4,2\} $
Rectified
$ t_1 \{4,2\} $
Birectified
$ t_2 \{4,2\} $
Truncated
$ t_{0,1} \{4,2\} $
Bitruncated
$ t_{1,2} \{4,2\} $
Cantellated
$ t_{0,2} \{4,2\} $
Cantitruncated
$ t_{0,1,2} \{4,2\} $
Square dihedron Square dihedron Square hosohedron Truncated square dihedron Cube Cube Octagonal prism
Regular
$ t_0 \{2,2\} $
Rectified
$ t_1 \{2,2\} $
Birectified
$ t_2 \{2,2\} $
Truncated
$ t_{0,1} \{2,2\} $
Bitruncated
$ t_{1,2} \{2,2\} $
Cantellated
$ t_{0,2} \{2,2\} $
Cantitruncated
$ t_{0,1,2} \{2,2\} $
Digonal dihedron Digonal dihedron Digonal dihedron Digonal prism Digonal prism Digonal prism Cube