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.
Hypervolumes[]
- vertex count = \(8\)
- edge length = \(12l\)
- 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
\()(\) |
Point
\(()\) |
Line segment
\(\{\}\) |
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 |