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Regular Hexahedron
Cube
(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
Coxeter-Dynkin CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
Symmetry Oh or (*432)
References U06, C18, W3
Properties Regular convex zonohedron
Dihedral angle 90°
Cube
4.4.4
(Vertex figure)
Octahedron.png
Octahedron
(dual polyhedron)
Cube
Net

In geometry, a cube[1] is a three-dimensional 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 three-dimensional 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 \sqrt{2}.

Contents

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 (x0x1x2) with −1 < x i < 1.

Formulae

For a cube of edge length a,

surface area 6a2
volume a3
face diagonal \sqrt 2a
space diagonal \sqrt 3a
radius of circumscribed sphere \frac{\sqrt 3}{2} a
radius of sphere tangent to edges \frac{a}{\sqrt 2}
radius of inscribed sphere \frac{a}{2}
angles between faces \frac{\pi}{2}

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).

Uniform colorings and symmetry

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 vertex-transitive coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h 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
Coxeter-Dynkin CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png CDW ring.pngCDW 4.pngCDW dot.pngCDW 2.pngCDW ring.png CDW ring.pngCDW 2.pngCDW ring.pngCDW 2.pngCDW ring.png
Schläfli symbol {4,3} {4}x{} {}x{}x{}
Wythoff symbol 3 | 4 2 4 2 | 2 | 2 2 2
Symmetry Oh
(*432)
D4h
(*422)
D2h
(*222)
D3d
(2*3)
Symmetry order 24 16 8 12
Image
(uniform coloring)
Hexahedron.png
(111)
Tetragonal prism.png
(112)
Uniform polyhedron 222-t012.png
(123)
Trigonal trapezohedron.png

Geometric relations

These familiar six-sided dice are cube-shaped.

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.

Other dimensions

The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or (rarely) hypercube.

The analogue of the cube in n-dimensional Euclidean space is called a hypercube or n-dimensional cube or simply n-cube. 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.

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 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.

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 edge-lengths? Equal angles? Right angles?
Cube Yes Yes Yes
Rhombohedron Yes Yes No
Cuboid No Yes Yes
Parallelepiped No Yes No
quadrilaterally-faced hexahedron No No No

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.

An extension is the 3-dimensional k-ary Hamming graph, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers.

See also

References

  1. ^ English cube from Old French < Latin cubus < Greek kubos, "a cube, a die, vertebra". In turn from PIE *keu(b)-, "to bend, turn".
  2. ^ * Weisstein, Eric W., "Cube" from MathWorld.

External links

This audio file was created from a revision dated 2006-07-07, and does not reflect subsequent edits to the article. (Audio help)
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Quotes

Up to date as of January 14, 2010
(Redirected to Cube (film) article)

From Wikiquote

Cube is a 1997 science-fiction film about several strangers trapped inside a maze of cube-shaped rooms.

Directed by Vincenzo Natali. Written by André Bijelic, Vincenzo Natali, and Graeme Manson.
Don't Look For A Reason…Look For A Way Out taglines

Contents

Leaven

  • My parents are these people, I live with them…I'm boring.

Worth

  • I'm just a guy. I work in an office building, doing office building stuff.
  • I don't wanna die, I'm just being realistic. You think they'd go to all the trouble to build this thing if we could just walk out?
  • Do you think we matter? We don't.
  • I mean, this is an accident, a forgotten, perpetual public works project. Do you think anybody wants to ask questions? All they want is a clear conscience and a fat paycheck.
  • I mean, nobody wants to see the big picture. Life's too complicated.
  • Just out of curiosity—I mean, don't hit me again, I think—but what are you going to do when you get there?

Quentin

  • Let's rule out aliens for now and concentrate on what we know.
  • You can't see the big picture from in here, so don't try. Keep your head down, keep it simple. Just look at what's in front of you.
  • Leaven…you beautiful brain.
  • What do you think the establishment is? It's just guys like me. Their desks are bigger but their jobs aren't. They don't conspire, they buy boats.
  • Every day I mop up after your bleeding heart. The only reason you even exist is because I keep you!
  • I know your type. No kids, no man to fuck you. So you go around outraged, sticking your nose up other people's assholes, sniffing their business!
  • I looked through the walls. I dreamed him, at his desk, designing everything. He can't let you solve the puzzle, see, because there is a purpose.
  • I'm not dying in a fucking rat maze!

Holloway

  • Only the government could build something this ugly.
  • It's okay, I just swallowed my button.
  • It's all the same machine, right? The Pentagon, multinational corporations, the police! You do one little job, you build a widget in Saskatoon and the next thing you know it's two miles under the desert, the essential component of a death machine!
  • God help you, Quentin. Did you smack your kids around too?

Rennes

  • Look around. Take a good, long look-see. 'Cause I got a feeling it's looking at us.
  • No more talkin'. No more guessin'. Don't even think about nothing that's not right in front of ya. That's the real challenge. You've got save yourselves from yourselves.
  • Suck on it...keeps the saliva flowin'.
  • Merde.

Dialogue

Worth: I have nothing to live for out there.
Leaven: What is out there?
Worth: Boundless human stupidity.

Quentin: Listen, we can't go climbing around in here.
Holloway: Why not?
Quentin: There's traps.
Holloway: What do you mean, traps?
Quentin: Traps. I looked in the room down there and something almost cut my head off.

Leaven: Shouldn't we wait here?
Holloway: For what?
Leaven: To see if anybody comes.
Worth: No one's going to come.

Quentin: This guy's the Wren.
Holloway: The what?
Quentin: He's the wren. The bird of Attica. Flew the coop on six major prisons.
Rennes: Seven.

Quentin: Don't you have a wife, or a girlfriend, or something?
Worth: Nope. I've got a pretty fine collection of pornography.

Kazan: This room is…green.
Holloway: Yes it is.
Kazan: I want to go back to the blue room.

Quentin: You could try and help me out here, buddy.
Worth: No I couldn't.

Holloway: I'm talking about where do you hide something this big. Hey, I'm sorry to shake your foundations, Quentin, but you have no idea where your tax dollars go.
Quentin: Free clinic doctors?
Holloway: Only the military-industrial complex could afford to build something this size.
Quentin: Holloway? What is the military-industrial complex? Have you ever been there? I'm telling you, it's not that complex.

Quentin: That your two bits worth…Worth?
Worth: For what it's worth.

Quentin: Somebody has to take responsibility around here.
Worth: And that somebody has to be you.
Quentin: Not all of us have the luxury of playing nihilist.
Worth: Not all of us are conceited enough to play hero.

Worth: Holloway, you don't get it.
Holloway: Then help me, please. I need to know.
Worth: This may be hard for you to understand, but there is no conspiracy. Nobody is in charge. It's a headless blunder operating under the illusion of a master plan. Can you grasp that? Big Brother is not watching you.

Quentin: Why put people in it?
Worth: Because it's here. You have to use it, or you admit it's pointless.
Quentin: But it is pointless!
Worth: Quentin, that's my point.

Leaven: If this were right, then we would be outside of the cube.
[Leaven looks around.]
Leaven: No, not outside of the cube.
Quentin: Oh. Guess that means we're not having dinner.

Holloway: How long did you know people were being put in here?
Worth: A couple of months.
Holloway: That's not long…if you consider your whole life.
Worth: I am.

Leaven: We'll come back for him...
Holloway: That's a lie, and you know it.

Quentin: Sure you can. It's your gift.
Leaven: It's not a gift. It's just a brain.

Leaven: This room moves to 0, 1, and -1 on the X-axis, 2, 5, and -7 on the Y and 1, -1, and 0 on zed.
Quentin: And what does that mean?
Leaven: You suck at math.

Leaven: What is out there?
Worth: Boundless human stupidity…

Holloway: It's so much worse than I thought.
Worth: Not really, just more pathetic.

Leaven: With those numbers, the room should be safe.
Quentin: [Laughing] Only one way to find out! [Calmly throws Worth into the room]

Taglines

  • Fear…Paranoia…Suspicion…Desperation
  • Don't Look For A Reason…Look For A Way Out
  • The Walls Are Closing In.

Cast

See also

External links

Wikipedia
Wikipedia has an article about:

1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

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 all-important 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 right-angled 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 (836-901) 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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Simple English

[[File:|thumb|A 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.

Relative 2-dimensional shape

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 2-dimensional 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.

Volume

  • The volume of a cube is the length of any one of the edges (they are all the same length so it does not matter which edge is used) cubed.
  • This means you multiply the number by itself, and then by itself again.
  • If the edge is named 'd' (See Diagram), the equation would be this: Volume=d×d×d (or Volume=d3).
File:Halite(Salt)
A magnified crystal of salt

Cubic shaped things








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