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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 , meaning that it is made of squares, three of which meet at each vertex. It can also be represented by the Schläfli symbols as it is the product of three line segments, as it is the product of a square and a line segment (in other words, a square-based prism) and 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:

### - 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.

### - 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 =
• surface area =
• surcell volume =(when a=b, this becomes the regular cube.)

### - 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 =
• surface area =
• surcell volume = (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.

### Subfacets

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

### 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 .

### Notations

• Toratopic notation:
• Tapertopic notation:

## Coordinate System

The coordinate system corresponding to the cube is called realm cartesian coordinates, with the three coordinates being . The length elements of cartesian coordinates are simply , and , giving a line element of with a length . The surface elements, giving the changes in area for small changes in x, y and z, are , and . The volume element, giving the change in volume for small changes in x, y and z, is .

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

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

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

Null polytope

Point

Line segment

Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Tridecahendon

... Omegasimplex
Cross

Square

Octahedron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

... Omegaball

Regular
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
Cube Cuboctahedron Octahedron Truncated cube Truncated octahedron Rhombicuboctahedron Great rhombicuboctahedron
Regular
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
Square dihedron Square dihedron Square hosohedron Truncated square dihedron Cube Cube Octagonal prism
Regular
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
Digonal dihedron Digonal dihedron Digonal dihedron Digonal prism Digonal prism Digonal prism Cube