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 
CoxeterDynkin  
Symmetry  O_{h} or (*432) 
References  U_{06}, C_{18}, W_{3} 
Properties  Regular convex zonohedron 
Dihedral angle  90° 
4.4.4 (Vertex figure) 
Octahedron (dual polyhedron) 
Net 
In geometry, a cube^{[1]} is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry).
A cube is the threedimensional case of the more general concept of a hypercube.
It has 11 nets.^{[2]} If one were to colour the cube so that no two adjacent faces had the same colour, one would need 3 colours.
If the original cube has edge length 1, its dual octahedron has edge length .
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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
while the interior consists of all points (x_{0}, x_{1}, x_{2}) with −1 < x_{ i} < 1.
For a cube of edge length a,
surface area  6a^{2} 
volume  a^{3} 
face diagonal  
space diagonal  
radius of circumscribed sphere  
radius of sphere tangent to edges  
radius of inscribed sphere  
angles between faces 
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.
A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).
The cube has 3 uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.
The cube has 3 classes of symmetry, which can be represented by vertextransitive coloring the faces. The highest octahedral symmetry O_{h} has all the faces the same color. The dihedral symmetry D_{4h} comes from the cube being a prism, with all four sides being the same color. 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.
Name  Regular hexahedron  Square prism  Cuboid  Trigonal trapezohedron 

CoxeterDynkin  
Schläfli symbol  {4,3}  {4}x{}  {}x{}x{}  
Wythoff symbol  3  4 2  4 2  2   2 2 2  
Symmetry  O_{h} (*432) 
D_{4h} (*422) 
D_{2h} (*222) 
D_{3d} (2*3) 
Symmetry order  24  16  8  12 
Image (uniform coloring) 
(111) 
(112) 
(123) 
The cube is unique among the Platonic solids for being able to tile Euclidean space regularly. 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).
The cube can be cut into 6 identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained.
The analogue of a cube in fourdimensional Euclidean space has a special name—a tesseract or (rarely) hypercube.
The analogue of the cube in ndimensional Euclidean space is called a hypercube or ndimensional cube or simply ncube. 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.
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 ⅓ of that of the cube. The remaining space consists of four equal irregular tetrahedra 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.
If two opposite corners of a cube are truncated at the depth of the 3 vertices directly connected to them, an irregular octahedron is obtained. Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron.
Two tetrahedra in the cube (stella octangula) 
The rectified cube (cuboctahedron) 
Truncated cube 
Cantellated cube (rhombicuboctahedron) 
Omnitruncated cube (truncated cuboctahedron) 
Snub cube 

An alternately truncated cube 
All but the last of the figures shown have the same symmetries as the cube (see octahedral symmetry).
The cube is a special case in various classes of general polyhedra:
Name  Equal edgelengths?  Equal angles?  Right angles? 

Cube  Yes  Yes  Yes 
Rhombohedron  Yes  Yes  No 
Cuboid  No  Yes  Yes 
Parallelepiped  No  Yes  No 
quadrilaterallyfaced hexahedron  No  No  No 
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.
An extension is the 3dimensional kary Hamming graph, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers.


Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
Cube is a 1997 sciencefiction film about several strangers trapped inside a maze of cubeshaped rooms.
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CUBE (Gr. Kb(30s, a cube), in geometry, a solid bounded by six equal squares, so placed that the angle between any pair of adjacent faces is a right angle. This solid played an allimportant part in the geometry and cosmology of the Greeks. Plato (Timaeus) described the figure in the following terms:  "The isosceles triangle which has its vertical angle a right angle. .. combined in sets of four, with the right angles meeting at the centre, form a single square. Six of these squares joined together formed eight solid angles, each produced by three plane right angles: and the shape of the body thus formed was cubical, having six square planes for its surfaces." In his cosmology Plato assigned this solid to "earth," for "` earth ' is the least mobile of the four (elements  ' fire,' ` water,' ` air ' and ` earth ') and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most stable." The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article Crystallography for its significance as a crystal form.
A famous problem concerning the cube, namely, to construct a cube of twice the volume of a given cube, was attacked with great vigour by the Pythagoreans, Sophists and Platonists. It became known as the "Delian problem" or the "problem of the duplication of the cube," and ranks in historical importance with the problems of "trisecting an angle" and "squaring the circle." The origin of the problem is open to conjecture. The Pythagorean discovery of "squaring a square," i.e. constructing a square of twice the area of a given square (which follows as a corollary to the Pythagorean property of a rightangled triangle, viz. the square of the hypotenuse equals the sum of the squares on the sides), may have suggested the strictly analogous problem of doubling a cube. Eratosthenes (c. 200 B.C.), however, gives a picturesque origin to the problem. In a letter to Ptolemy Euergetes he narrates the history of the problem. The Delians, suffering a dire pestilence, consulted their oracles, and were ordered to double the volume of the altar to their tutelary god, Apollo. An altar was built having an edge double the length of the original; but the plague was unabated, the oracles not having been obeyed. The error was discovered, and the Delians applied to Plato for his advice, and Plato referred them to Eudoxus. This story is mere fable, for the problem is far older than Plato.
Hippocrates of Chios (c. 430 B.C.), the discoverer of the square of a lune, showed that the problem reduced to the determination of two mean proportionals between two given lines, one of them being twice the length of the other. Algebraically expressed, if x and y be the required mean proportionals and a, 2a, the lines, we have a: x :: x: y :: y : 2a, from which it follows that x = 2a3. Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of algebraic geometry, is to solve geometrically a cubic equation. According to Proclus, a man named Hippias, probably Hippias of Elis (c. 460 B.C.), trisected an angle with a mechanical curve, named the quadratrix. Archytas of Tarentum (c. 430 B.C.) solved the problems by means of sections of a half cylinder; according to Eutocius, Menaechmus solved them by means of the intersections of conic sections; and Eudoxus also gave a solution.
All these solutions were condemned by Plato on the ground that they were mechanical and not geometrical, i.e. they were not effected by means of circles and lines. However, no proper geometrical solution, in Plato's sense, was obtained; in fact it is now generally agreed that, with such a restriction, the problem is insoluble. The pursuit of mechanical methods furnished a stimulus to the study of mechanical loci, for example, the locus of a point carried on a rod which is caused to move according to a definite rule. Thus Nicomedes invented the conchoid; Diodes the cissoid; Dinostratus studied the quadratrix invented by Hippias; all these curves furnished solutions, as is also the case with the trisectrix, a special form of Pascal's limacon. These problems were also attacked by the Arabian mathematicians; Tobit ben Korra (836901) is credited with a solution, while Abul Gud solved it by means of a parabola and an equilateral hyperbola.
In algebra, the "cube" of a quantity is the quantity multiplied by itself twice, i.e. if a be the quantity a X a X a(= a ) is its cube. Similarly the "cube root" of a quantity is another quantity which when multiplied by itself twice gives the original quantity; thus is the cube root of a (see Arithmetic and Algebra). A "cubic equation" is one in which the highest power of the unknown is the cube (see Equation); similarly, a "cubic curve" has an equation containing no term of a power higher than the third, the powers of a compound term being added together.
In mensuration, "cubature" is sometimes used to denote the volume of a solid; the word is parallel with "quadrature," to determine the area of a surface (see Mensuration; Infinitesimal Calculus) .
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Categories: CROCUN  Mathematics
[[File:thumbA cube has 6 sides of equal length and width]] A cube is a block with all right angles and whose height, width and depth are all the same.
A cube is one of the simplest mathematical shapes in space. Something that is shaped like a cube is sometimes referred to as cubic.
The basic difference in cube and a square is, cube is 3d figure(having 3 dimensions i.e.length,breadth and height while square have only 2 dimensions i.e length and breadth. The 2dimensional shape (like a circle, square, triangle, etc) that a cube is made of is squares. The sides (faces) of a cube are squares. The edges are straight lines. The corners (vertices) are at right angles. A cube has 8 corners, 12 edges and 6 sides.example dice having six number denotes 6 faces.
