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In geometry, a cube[a] is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross.[1]
The cube is also a square parallelepiped, an equilateral cuboid, a right rhombohedron, and a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
The cube is dual to the octahedron. It has cubical or octahedral symmetry, and is the only convex polyhedron whose faces are all squares. Its generalization for higher-dimensional spaces is called a hypercube.
The cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
This configuration matrix represents the cube. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[2] For example, the 2 in the first column of the middle row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 3 in the middle column of the first row indicates that 3 edges meet at each vertex.
Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube. They were unable to solve this problem, which in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.
The cube has four 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 solid, with all the six sides being different colors. The prismatic subsets D2d has the same coloring as the previous one and D2h has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.
A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges.[4] To color the cube so that no two adjacent faces have the same color, one would need at least three colors.
The cube is the cell of the only regular tiling of three-dimensional Euclidean space. 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 six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces).
Cubes appear in Abrahamic religions. The Kaaba (Arabic for 'cube') in Mecca is one example. Cubes also appear in Judaism as tefillin, and the New Jerusalem is described in the New Testament as a cube.[5]
The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron; more generally this is referred to as a demicube. 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.
The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with six octagonal faces and eight 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 three 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.
The skeleton of the cube (the vertices and edges) forms a graph with 8 vertices and 12 edges, called the cube graph. It is a special case of the hypercube graph.[6] It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.
Cube is a 1997 Canadian science fiction horror-thriller film directed and co-written by Vincenzo Natali.[8] A product of the Canadian Film Centre's First Feature Project,[9] Nicole de Boer, Nicky Guadagni, David Hewlett, Andrew Miller, Julian Richings, Wayne Robson, and Maurice Dean Wint star as individuals trapped in a bizarre and deadly labyrinth of cube-shaped rooms.
Cube gained notoriety and a cult following for its surreal and Kafkaesque setting in industrial, cube-shaped rooms. It received generally positive reviews and led to a series of films. An American remake, currently on hold, was in development at Lionsgate in 2015,[10] though current development is unknown. A Japanese remake was released in 2021.
A man named Alderson awakens in a mysterious cube-shaped room which has hatches on each wall, ceiling and floor. Each hatch leads to similar interconnected rooms. He enters another room, but is killed by a razor wire trap.
Five different people, Quentin, Holloway, Worth, Leaven and Rennes, all meet in another room, unaware of how or why they are there. Quentin, a divorced police officer, warns the group that he has seen booby-traps in some of the other rooms. Leaven, a young mathematics student, notices each hatch has metal plates with three sets of numbers etched into them. Rennes, an escape artist who has fled seven prisons, tests his theory that each trap could be set by motion detectors by throwing his boot into a room. This initially works, but after jumping into a room, he is killed by acid. The group, horrified, realizes each trap is set by different sensors.
Worth's knowledge of the outer shell's dimensions allows Leaven to calculate that with 26 rooms in each row, the entire Cube has 17,576 rooms, plus an additional 'bridge' room that would connect to the outer shell. She realizes that the numbers may indicate each room's Cartesian coordinates. Following the theory, the group travels to the outer edge but realize every room there is trapped. Rather than backtrack, they traverse a room with a sound-activated trap. After Kazan nearly causes Quentin's death by accident, Holloway defends him from Quentin's threats, insinuating that Quentin may be an abusive husband who likes young girls.
The group reaches the edge, finding a bottomless abyss separating the Cube from the outer shell. Being one of the lightest, Holloway tries to swing over to the outer shell, lowered down using a rope made of the group's uniforms tied together. The Cube shakes, causing everyone to accidentally release the rope, and Quentin catches it at the last second. He initially pulls her up, but then lets her fall to her death.
Quentin, becoming more unhinged, persuades Leaven to abandon Kazan and Worth. He tries to sexually assault her, but Worth attacks him. Quentin counters savagely, dropping him into the room below. Worth starts laughing hysterically, realizing they are in the same room Rennes died, indicating they have been traveling in circles. Quentin is horrified, but Worth finds the room where Rennes died in has now moved to the edge of the maze. Leaven deduces that traps are not tagged by prime numbers, but by powers of prime numbers. Kazan is revealed as an autistic savant[11] who can mentally calculate prime factorizations. With this newfound knowledge, Leaven guides the group to the edge cube, using Kazan's calculations. Worth then traps Quentin in a hatch, letting Leaven and Kazan escape from him. When Quentin finds them, he attempts to harm them, before Worth opens the hatch under him from the room below. The others travel to the bridge room where they open the exit hatch, seeing a bright light.
As Leaven attempts to persuade the guilt-stricken Worth, who no longer wishes to escape, Quentin reappears and impales her with a hatch lever. Worth angrily attacks Quentin. Quentin heavily wounds him in the struggle and pursues Kazan to the other side. Worth grabs Quentin's legs, pinning him in between the hatch. The cubes move, splitting him in half. Worth, bleeding badly, crawls to Leaven who lays near unmoving to stay by her side. Kazan wanders out into the bright light, his fate left unknown.
On casting Maurice Dean Wint as Quentin, Natali's cost-centric approach sought an actor for a split-personality role of hero and villain. Wint was considered the standout among the cast and was confident that the film would be a breakthrough for the Canadian Film Centre.[13]
Director Vincenzo Natali did not have confidence in financing a film. He cost-reduced his pitch with a single set reused as many times as possible, with the actors moving around a virtual maze.[15] As the most expensive element, a set with a cube and a half was built off the floor, to allow the surroundings to be lit from behind all walls of the cube.[16] In 1990, Natali had had the idea to make a film "set entirely in hell", but in 1994 while working as a storyboard artist's assistant at Canada's Nelvana animation studio, he completed the first script for Cube. The initial draft had a more comedic tone, surreal images, a cannibal, edible moss growing on the walls, and a monster that roamed the Cube. Roommate and childhood filmmaking partner Andre Bijelic helped Natali strip the central idea to its essence of people avoiding deadly traps in a maze. Scenes outside the cube were deleted, and the identity of the victims changed. In some drafts, they were accountants and in others criminals, with the implication that their banishment to the Cube was part of a penal sentence. One of the most important dramatic changes was the removal of food and water for a more urgent escape.[17]
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