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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

$\{4,3\}$ , 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.

### Subfacets

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

• 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:$111$

## 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

$dx dy dz$ .

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

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

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

$\{3^9\}$

$\{3^{10}\}$

Tridecahendon

$\{3^{11}\}$

$\{3^{12}\}$

$\{3^{13}\}$

$\{3^{14}\}$

$\{3^{15}\}$

... Omegasimplex
Cross

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

Square

$\{4\}$

Octahedron

$\{3, 4\}$

$\{3^2, 4\}$

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

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

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

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

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

... Omegacross
Hydrotopes

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

Pentagon

$\{5\}$

Icosahedron

$\{3, 5\}$

Hexacosichoron

$\{3^2, 5\}$

Order-5 pentachoric tetracomb

$\{3^3, 5\}$

Order-5 hexateric pentacomb

$\{3^4, 5\}$

...
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

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

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

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

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

... Omegeract
Cosmotopes

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

Pentagon

$\{5\}$

Dodecahedron

$\{5, 3\}$

Hecatonicosachoron

$\{5, 3^2\}$

Order-3 hecatonicosachoric tetracomb

$\{5, 3^3\}$

Order-3-3 hecatonicosachoric pentacomb

$\{5, 3^4\}$

...
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

$\mathbb B^{13}$

$\mathbb B^{14}$

$\mathbb B^{15}$

$\mathbb B^{16}$

... Omegaball

$\mathbb B^{\aleph_0}$

$\{4,2\}$ $\{4,3\}$ $\{4,4\}$ $\{4,5\}$ $\{4,6\}$ $\{4,7\}$ $\{4,8\}$ ... $\{4,\infty\}$ $\{4, \frac{\pi i}{\lambda}\}$
Square dihedron Cube Square tiling Order-5 square tiling Order-6 square tiling Order-7 square tiling Order-8 square tiling ... Infinite-order square tiling Imaginary-order square tiling
$\{2,3\}$ $\{3,3\}$ $\{4,3\}$ $\{5,3\}$ $\{6,3\}$ $\{7,3\}$ $\{8,3\}$ ... $\{\infty,3\}$ $\{\frac{\pi i}{\lambda},3$
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling Order-3 pseudogonal tiling
${t}_{0,1} \{2,2\}$ ${t}_{0,1} \{2,3\}$ ${t}_{0,1} \{2,4\}$ ${t}_{0,1} \{2,5\}$ ${t}_{0,1} \{2,6\}$ ${t}_{0,1} \{2,7\}$ ${t}_{0,1} \{2,8\}$ ... ${t}_{0,1} \{2,\infty\}$ ${t}_{0,1} \{2,\frac{\pi i}{\lambda}$
Digonal prism Triangular prism Cube Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism ... Apeirogonal prism Pseudogonal 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