10-cube

In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

It can be named by its Schläfli symbol {4,3⁸}, being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the 4-cube) and deka- for ten (dimensions) in Greek, It can also be called an icosaxennon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates

Cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are

while the interior of the same consists of all points (x₀, x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉) with −1 < x_i < 1.

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

The cube is the only regular hexahedron and is one of the five Platonic solids and has 12 edges, 6 faces and 8 vertices.

The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. 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.

Orthogonal projections

The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A₂ and B₂Coxeter planes.

Spherical tiling

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.

Borg (Star Trek)

The Borg are a fictional alien race that appear as recurring antagonists in the Star Trek franchise. The Borg are a collection of species that have been turned into cybernetic organisms functioning as drones in a hive mind called "the Collective" or "the Hive". The Borg use a process called assimilation to force other species into the Collective by violent injection of microscopic machines called nanoprobes. The Borg's ultimate goal is "achieving perfection". Aside from being the main threat in First Contact, the Borg play major roles in The Next Generation and Voyager television series, primarily as an invasion threat to the United Federation of Planets, and serve as the way home to the Alpha Quadrant for isolated Federation starship Voyager.

The Borg have become a symbol in popular culture for any juggernaut against which "resistance is futile". TV Guide named the Borg #4 in their 2013 list of the 60 Nastiest Villains of All Time.

Concept

In the text commentary to the Collector's Edition of Star Trek: First Contact, Michael Okuda revealed that Star Trek: The Next Generation (TNG) writers began to develop the idea of the Borg as early as the first season episode, "Conspiracy", which introduced a coercive, symbiotic life form that took over key Federation personnel. It was thwarted by the Enterprise crew and presumably never heard of again (the 'alien conspiracy' plotline itself was scrapped when it became clear that the concept was too grim for Star Trek's target audience). Plans to feature the Borg as an increasingly menacing threat were subsequently scrapped in favor of a more subtle introduction, beginning with the mystery of missing colonies on both sides of the Neutral Zone in "The Neutral Zone" and culminating in the encounter between Borg and the Enterprise crew in "Q Who?".

Cube (disambiguation)

A cube is any regular, six-sided, three-dimensional solid object.

Cube may also refer to:

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