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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.

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.

## 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.

## 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 $dx dy dz$.

Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
Hypercube Point Line segment Square Cube Tesseract Penteract Hexeract Hepteract Octeract Enneract Dekeract Hendekeract Dodekeract Tridekeract Tetradekeract Pentadekeract Hexadekeract ... Omegeract
Hyperball Point Line segment Disk Ball Gongol Pentorb Hexorb Heptorb Octorb Enneorb Dekorb Hendekorb Dodekorb Tridekorb Tetradekorb Pentadekorb Hexadekorb ... Omegaball
$\{4,2\}$ $\{4,3\}$ $\{4,4\}$ $\{4,5\}$ $\{4,6\}$ ... $\{4,\aleph_0\}$
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