The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x0, x1) with -1 < xi < 1.
In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. Taking the square is exponentiation with exponent two. If x is a positive real number, the value of x2 is equal to the area of a (geometric) square of edge length x. If y = x2 and x≥0, then x is the square root of y.
In urban planning, a square is a planned open area in a city, usually or originally square in shape. See Red Square, Tiananmen Square, Times Square and Trafalgar Square for examples. Some are permanently or on certain days of the week used for an open market.
In carpentry, a square is a guide for establishing right angles (ninety-degree angles), usually made of metal and in the shape of a right triangle.
In the geometry narrative Flatland, A Square is the name of the main narrator of the story.
In video games, Square is short for Squaresoft.
In the parlance of jazz, a square was a person who failed to appreciate the medium, i.e. one who was out of date or out of touch. The term has persisted and permeated into mainstream cultures, as exemplified in Elvis Costello's hit Hip to be square, which, in ultimate self-reference, was used by Sesame Street to illustrate the geometrical meaning of "square".