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
}

{\displaystyle \{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

{\displaystyle { \{ \} }^{ 3 }}

 as it is the product of three line segments,

{
4
}
×
{
}

{\displaystyle \{ 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
}

{\displaystyle 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
}

{\displaystyle \{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
}
×
{
}

{\displaystyle \{ 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

(

2
a
b

)

{\displaystyle 4 \left( 2a + b \right) }

* surface area =

2
a

(

a
2
b

)

{\displaystyle 2a \left( a + 2b \right) }

* surcell volume =

a
2

b

{\displaystyle a^2b}

(when a=b, this becomes the regular cube.)

###

{

}
3

{\displaystyle \{\}^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

(

a
b
c

)

{\displaystyle 4\left(a + b + c\right)}

* surface area =

2

(

a
b
a
c
b
c

)

{\displaystyle 2\left(ab + ac + bc\right) }

* surcell volume =

a
b
c

{\displaystyle 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

{\displaystyle 8}

* edge length =

12
l

{\displaystyle 12 l}

* surface area =

6

l
2

{\displaystyle 6l^2}

* surcell volume =

l
3

{\displaystyle 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:

3

2

l

{\displaystyle \frac{\sqrt{3}}{2}l}

* Edge radius:

2

2

l

{\displaystyle \frac{\sqrt{2}}{2}l}

* Face radius:

1

/

2
l

{\displaystyle 1/2l}

### 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

{\displaystyle \max(x^2,y^2,z^2) = 1}

.

### Notations

* Toratopic notation:

{\displaystyle |||}

* Tapertopic notation:

111

{\displaystyle 111}

### Related shapes

* Dual: Octahedron
* Vertex figure: Equilateral triangle, side length

2

{\displaystyle \sqrt{2}}

## Coordinate System

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

(

x
,
y
,
z

)

{\displaystyle \left(x, y, z\right)}

. The length elements of cartesian coordinates are simply

d
x

{\displaystyle dx}

,

d
y

{\displaystyle dy}

 and

d
z

{\displaystyle dz}

, giving a line element of

d
# s
d
x

x
^

d
y

y
^

d
z

z
^

{\displaystyle ds = dx \hat{x} + dy \hat{y} + dz \hat{z}}

 with a length

d

s
2

=
d

x
2

d

y
2

d

z
2

{\displaystyle 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

d
x
d
y

{\displaystyle dx dy}

,

d
x
d
z

{\displaystyle dx dz}

 and

d
y
d
z

{\displaystyle dy dz}

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

d
x
d
y
d
z

{\displaystyle dx dy dz}

.

## See Also

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

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

{
4
,
2
}

{\displaystyle \{4,2\}}

{
4
,
3
}

{\displaystyle \{4,3\}}

{
4
,
4
}

{\displaystyle \{4,4\}}

{
4
,
5
}

{\displaystyle \{4,5\}}

{
4
,
6
}

{\displaystyle \{4,6\}}

{
4
,
7
}

{\displaystyle \{4,7\}}

{
4
,
8
}

{\displaystyle \{4,8\}}

{
4
,
∞
}

{\displaystyle \{4,\infty\}}

{
4
,

π
i

λ

}

{\displaystyle \{4, \frac{\pi i}{\lambda}\}}

{
2
,
3
}

{\displaystyle \{2,3\}}

{
3
,
3
}

{\displaystyle \{3, 3\}}

{
4
,
3
}

{\displaystyle \{4,3\}}

{
5
,
3
}

{\displaystyle \{5, 3\}}

{
6
,
3
}

{\displaystyle \{6,3\}}

{
7
,
3
}

{\displaystyle \{7,3\}}

{
8
,
3
}

{\displaystyle \{8,3\}}

{
∞
,
3
}

{\displaystyle \{\infty,3\}}

{

π
i

λ

,
3

{\displaystyle \{\frac{\pi i}{\lambda},3}

t

0
,
1

{
2
,
2
}

{\displaystyle {t}\_{0,1} \{2,2\}}

t

0
,
1

{
2
,
3
}

{\displaystyle {t}\_{0,1} \{2,3\}}

t

0
,
1

{
2
,
4
}

{\displaystyle {t}\_{0,1} \{2,4\}}

t

0
,
1

{
2
,
5
}

{\displaystyle {t}\_{0,1} \{2,5\}}

t

0
,
1

{
2
,
6
}

{\displaystyle {t}\_{0,1} \{2,6\}}

t

0
,
1

{
2
,
7
}

{\displaystyle {t}\_{0,1} \{2,7\}}

t

0
,
1

{
2
,
8
}

{\displaystyle {t}\_{0,1} \{2,8\}}

t

0
,
1

{
2
,
∞
}

{\displaystyle {t}\_{0,1} \{2,\infty\}}

t

0
,
1

{
2
,

π
i

λ

}

{\displaystyle {t}\_{0,1} \{2,\frac{\pi i}{\lambda}\}}

t
0

{
4
,
3
}

{\displaystyle t\_0 \{4,3\}}

t
1

{
4
,
3
}

{\displaystyle t\_1 \{4,3\}}

t
2

{
4
,
3
}

{\displaystyle t\_2 \{4,3\}}

t

0
,
1

{
4
,
3
}

{\displaystyle t\_{0,1} \{4,3\}}

t

1
,
2

{
4
,
3
}

{\displaystyle t\_{1,2} \{4,3\}}

t

0
,
2

{
4
,
3
}

{\displaystyle t\_{0,2} \{4,3\}}

t

0
,
1
,
2

{
4
,
3
}

{\displaystyle t\_{0,1,2} \{4,3\}}

t
0

{
4
,
2
}

{\displaystyle t\_0 \{4,2\}}

t
1

{
4
,
2
}

{\displaystyle t\_1 \{4,2\}}

t
2

{
4
,
2
}

{\displaystyle t\_2 \{4,2\}}

t

0
,
1

{
4
,
2
}

{\displaystyle t\_{0,1}\{4,2\}}

t

1
,
2

{
4
,
2
}

{\displaystyle t\_{1,2} \{4,2\}}

t

0
,
2

{
4
,
2
}

{\displaystyle t\_{0,2} \{4,2\}}

t

0
,
1
,
2

{
4
,
2
}

{\displaystyle t\_{0,1,2} \{4,2\}}

t
0

{
2
,
2
}

{\displaystyle t\_0 \{2,2\}}

t
1

{
2
,
2
}

{\displaystyle t\_1 \{2,2\}}

t
2

{
2
,
2
}

{\displaystyle t\_2 \{2,2\}}

t

0
,
1

{
2
,
2
}

{\displaystyle t\_{0,1} \{2,2\}}

t

1
,
2

{
2
,
2
}

{\displaystyle t\_{1,2} \{2,2\}}

t

0
,
2

{
2
,
2
}

{\displaystyle t\_{0,2} \{2,2\}}

t

0
,
1
,
2

{
2
,
2
}

{\displaystyle t\_{0,1,2} \{2,2\}}