Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n-dimensions is equal to .

An n-dimensional hypercube is also called an n-cube or an n-dimensional cube. The term "measure polytope" is also used, notably in the work of H. S. M. Coxeter (originally from Elte, 1912), but it has now been superseded.

The hypercube is the special case of a hyperrectangle (also called an n-orthotope).

A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2ⁿ points in Rⁿ with coordinates equal to 0 or 1 is called "the" unit hypercube.

Construction

This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the d-dimensional hypercube is the Minkowski sum of d mutually perpendicular unit-length line segments, and is therefore an example of a zonotope.

Cube 2: Hypercube

Cube 2: Hypercube is a 2002 Canadian science fiction psychological thriller film and the sequel to Cube. Released in 2002, Cube 2: Hypercube had a different director than its predecessor, Andrzej Sekuła. The industrial-style rooms of the first film are replaced with high-tech, brightly lit chambers; instead of traps such as flamethrowers and extending spikes, the rooms have "evolved" and now are controlling illusion, time, space and reality.

Plot

The film starts with a young woman named Becky (Greer Kent), shown to be trapped in the Cube. She enters another room, but unknown to her, the room has reversed gravity, as she is pulled up.

Some time later, a woman named Kate (Kari Matchett), detective Simon (Geraint Wyn Davies), a blind girl named Sasha (Grace Lynn Kung), engineer Jerry (Neil Crone), game developer Max (Matthew Ferguson), lawyer Julia (Lindsey Connell), and an elderly woman named Mrs. Paley (Barbara Gordon) find themselves trapped in brightly lit cubes, each with a panel on each of the six sides, which are doors to other rooms. They come across Colonel Thomas Maguire (Bruce Gray), who says that they have to solve the code in order to leave the mysterious place. A wall begins to close in on the group. The group escapes while Thomas stays behind. Kate and Simon watch in horror as Thomas is disintegrated by the wall. Later experiences around the cube reveal that gravity can operate in different directions in each room, while Mrs. Paley (revealed to be a retired theoretical mathematician) and Jerry realize that they may be in a tesseract, or a hypercube. Kate notices the numbers "60659" everywhere they go.

Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.

A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.

Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a parallelogram (opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles) and therefore has all the properties of all these shapes, namely: