Cross-polytope

In geometry, a cross-polytope,orthoplex,hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The n-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ₁-norm on Rⁿ:

In 1 dimension the cross-polytope is simply the line segment [−1, +1], in 2 dimensions it is a square (or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these.

The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T(2n,n).

Blow Your Mind

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Norma

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Project

In contemporary business and science a project is a collaborative enterprise, involving research or design, that is carefully planned to achieve a particular aim.

Project can also be defined as a set of interrelated tasks to be executed over a fixed period and within certain cost and other limitations.

Projects can be further defined as temporary rather than permanent social systems or work systems that are constituted by teams within or across organizations to accomplish particular tasks under time constraints. An ongoing project is usually called (or evolves into) a program.

Overview

The word project comes the Latin word projectum from the Latin verb proicere, "before an action" which in turn comes from pro-, which denotes precedence, something that comes before something else in time (paralleling the Greek πρό) and iacere, "to do". The word "project" thus actually originally meant "before an action".

When the English language initially adopted the word, it referred to a plan of something, not to the act of actually carrying this plan out. Something performed in accordance with a project became known as an "object".Every project has certain phases of development.