Cube

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 (D_4h) 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 = $a^2b$ (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[]

vertex count = $8$

edge length = $12 l$

surface area = $6l^2$

surcell volume = $l^3$

Subfacets[]

8 points (0D)
12 line segments (1D)
6 squares (2D)
1 cube (3D)

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[]

Dual: Octahedron
Vertex figure: Equilateral triangle, side length $\sqrt{2}$

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[]

Polytope Wiki. "Cube".
Bowers, Jonathan. "Polyhedron Category 1: Regulars".

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	Seventeen	...	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 $\{3^9\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	Octadecapedakon $\{3^{16}\}$	...	Omegasimplex
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 $\{3^{14}, 4\}$	Heptadekacross $\{3^{15},4\}$	...	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 $\{4, 3^{11}\}$	Tetradekeract $\{4, 3^{12}\}$	Pentadekeract $\{4, 3^{13}\}$	Hexadekeract $\{4, 3^{14}\}$	Heptadekeract $\{4,3^{15}\}$	...	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 $\mathbb B^{13}$	Tetradekorb $\mathbb B^{14}$	Pentadekorb $\mathbb B^{15}$	Hexadekorb $\mathbb B^{16}$	Heptadekorb $\mathbb {B} ^{17}$	...	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