This article is about the geometric shape. For other meanings of the word "cube", see cube (disambiguation).

Regular Hexahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F = 6, E = 12 V = 8 (χ = 2)
Faces by sides	6{4}
Schläfli symbol	{4,3}
Wythoff symbol	3 | 2 4 2 4 | 2 2 2 2 |
Coxeter-Dynkin
Symmetry	Oh
References	U06, C18, W3
Properties	Regular convex zonohedron
Dihedral angle	90°
4. In Geometry, a Platonic solid is a convex Regular polyhedron. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, the Schläfli symbol is a notation of the form {pqr In Geometry, a Wythoff symbol is a short-hand notation created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a List of Symmetry groups on the sphere Spherical symmetry groups are also called Point groups in three dimensions. A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the A zonohedron is a convex Polyhedron where every face is a Polygon with point Symmetry or equivalently symmetry under Rotations through In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. 4. 4 (Vertex figure)	Octahedron (dual polyhedron)
Net

A cube^[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off An octahedron (plural octahedra is a Polyhedron with eight faces In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the In Geometry the net of a Polyhedron is an arrangement of edge-joined Polygons in the plane which can be folded (along edges to become the faces of the polyhedron Three-dimensional space is a geometric model of the physical Universe in which we live Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides The cube can also be called a regular hexahedron and is one of the five Platonic solids. A hexahedron (plural hexahedra is a Polyhedron with six faces In Geometry, a Platonic solid is a convex Regular polyhedron. It is a special kind of square prism, of rectangular parallelepiped and of 3-sided trapezohedron. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron The cube is dual to the octahedron. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the An octahedron (plural octahedra is a Polyhedron with eight faces It has cubical symmetry (also called octahedral symmetry). A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation A cube is the three-dimensional case of the more general concept of a hypercube, which exists in any dimension. In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3

Cartesian coordinates

For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are

(±1,±1,±1)

while the interior consists of all points (x₀, x₁, x₂) with -1 < x_i < 1. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

Formulae

For a cube of edge length a,

surface area	6a2
volume	a3
radius of circumscribed sphere
radius of sphere tangent to edges
radius of inscribed sphere

As the volume of a cube is the third power of its sides a×a×a, third powers are called cubes, by analogy with squares and second powers. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Arithmetic and Algebra, the cube of a number n is its third power &mdash the result of multiplying it by itself three times In Arithmetic and Algebra, the cube of a number n is its third power &mdash the result of multiplying it by itself three times In Algebra, the square of a number is that number multiplied by itself

A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong Surface area is the measure of how much exposed Area an object has Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).

Symmetry

The cube has 3 classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry O_h has all the faces the same color. A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation The dihedral symmetry D_4h comes from the cube being a prism, with all four sides being the same color. This article deals with three infinite series of Point groups in three dimensions which have a Symmetry group which as abstract group is a Dihedral group Dih The lowest symmetry D_2h is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. In Geometry, a Wythoff symbol is a short-hand notation created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a

(3 colors) | 2 2 2 D2h

(2 colors) 4 2 | 2 D4h

(1 color) 3 | 4 2 Oh

Geometric relations

The familiar six-sided dice are cube shaped

The cube is unique among the Platonic solids for being able to tile space regularly. For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). A zonohedron is a convex Polyhedron where every face is a Polygon with point Symmetry or equivalently symmetry under Rotations through

Other dimensions

The analogue of a cube in four-dimensional Euclidean space has a special name — a tesseract or (rarely) hypercube. The 1967 International and Universal Exposition, or Expo 67 as it was commonly known was the World's Fair held in Montreal, Canada from April 27 to October Geometry The tesseract can be constructed in a number of different ways In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3

The analogue of the cube in n-dimensional Euclidean space is called a hypercube or n-dimensional cube or simply n-cube. In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3 It is also called a measure polytope.

There are analogues of the cube in lower dimensions too: a point in dimension 0, a segment in one dimension and a square in two dimensions. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides

Related polyhedra

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. These two together form a regular compound, the stella octangula. A polyhedral compound is a Polyhedron that is itself composed of several other polyhedra sharing a common centre See also Polyhedron Merkaba Polyhedron models * Plane (metaphysics The intersection of the two forms a regular octahedron. An octahedron (plural octahedra is a Polyhedron with eight faces 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 ⅓ 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. In Euclidean geometry, rectification is the process of truncating a Polytope by marking the midpoints of all its edges and cutting off its vertices at those points A cuboctahedron is a Polyhedron with eight triangular faces and six square faces If smaller corners are cut off we get a polyhedron with 6 octagonal faces and 8 triangular ones. Regular octagons A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size In particular we can get regular octagons (truncated cube). The truncated cube, or truncated hexahedron, is an Archimedean solid. The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount. The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces

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. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal A polyhedral compound is a Polyhedron that is itself composed of several other polyhedra sharing a common centre

The tetrahedra in the cube (stella octangula)	The rectified cube (cuboctahedron)	Truncated cube	Rhombicuboctahedron
Compound of three cubes	An alternately truncated cube

All but the last of the figures shown have the same symmetries as the cube (see octahedral symmetry). See also Polyhedron Merkaba Polyhedron models * Plane (metaphysics In Euclidean geometry, rectification is the process of truncating a Polytope by marking the midpoints of all its edges and cutting off its vertices at those points A cuboctahedron is a Polyhedron with eight triangular faces and six square faces The truncated cube, or truncated hexahedron, is an Archimedean solid. The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces Cartesian coordinates Cartesian coordinates for the vertices of this compound are all the Permutations of (±&radic2 0 ±1 A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation

Combinatorial cubes

A different kind of cube is the cube graph, which is the graph of vertices and edges of the geometrical cube. It is a special case of the hypercube graph. In the mathematical field of Graph theory, the Hypercube graph Qn is a Regular graph with 2 n

An extension is the 3-dimensional k-ary Hamming graph, which for k = 2 is the cube graph. Hamming graphs are a special class of graphs used in several branches of Mathematics and Computer science. Graphs of this sort occur in the theory of parallel processing in computers. Parallel computing is a form of computation in which many instructions are carried out simultaneously operating on the principle that large problems can often

See also

• Unit cube
• Kaaba, a large masonry structure roughly the shape of a cube

References

External links

• Eric W. Weisstein, Cube at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
• Cube: Interactive Polyhedron Model -- works right in your web browser
• The Uniform Polyhedra

• Virtual Reality Polyhedra
• Paper Models of Polyhedra
• Volume of a cube With interactive animation