Cube
From Freepedia
- This article is about the geometric shape. For other meanings of the word "cube", see cube (disambiguation).
| Cube | |
|---|---|
| Image:Hexahedron.jpg Click on picture for large version. Click here for spinning version. | |
| Type | Platonic |
| Face polygon | square |
| Faces | 6 |
| Edges | 12 |
| Vertices | 8 |
| Faces per vertex | 3 |
| Vertices per face | 4 |
| Symmetry group | octahedral (Oh) of order 48 |
| Dual polyhedron | octahedron |
| Properties | regular, convex, zonohedron |
| Image:Cube vertfig.png Vertex Figure | |
A cube (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. The cube is a special kind of square prism, of rectangular parallelepiped and of 3-sided trapezohedron, and is dual to the octahedron. Thus it has octahedral symmetry.
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Canonical coordinates
Canonical coordinates for the vertices of a cube centered at the origin are (±1,±1,±1), while the interior of the same consists of all points (x0, x1, x2) with -1 < xi < 1.
Area and volume
The area A and volume V of a cube of edge length a are:
- <math>A=6a^2</math>
- <math>V=a^3</math>
A cube construction has the largest volume among cuboids (rectangular boxes) with a given surface area (e.g., paper, cardboard, sheet metal, etc.). Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).
The cube is unique among the Platonic solids for being able to tile space regularly.
Higher dimensions
Image:Expo 67 cubes in a room.jpg
In the four-dimensional Euclidean space, the analogue of a cube has a special name — a tesseract or hypercube.
The analog of the cube in the n-dimensional Euclidean space is called n-dimensional cube, or simply cube, if it does not lead to a confusion. The name measure polytope is also used.
Related polyhedra
The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.
One such regular tetrahedron has a volume of 1/3 of that of the cube. The remaining space consists of four equal irregular polyhedra with a volume of 1/6 of that of the cube, each.
The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with 6 octagonal faces and 8 triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount.
A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes.
The figures shown have the same symmetries as the cube (see octahedral symmetry).
Trivia
If each edge of a cube is replaced by a one ohm resistor, the resistance between opposite vertices is 5/6 ohms, and that between adjacent vertices 7/12 ohms.
See also
External links
- The Uniform Polyhedra
- Spinning Hexahedron
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Paper Models of Polyhedra Many links
