Cube
- This article is about the geometric shape. For other meanings of the word "cube", see cube (disambiguation).
| Cube | |
|---|---|
| Cube 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 |
| Cube 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.
Contents
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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=6a^2}
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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
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.
Template:- 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
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