squares

A square is a regular quadrilateral with four equal sides and four right angles. Its diagonals are equal in length and cross at the center, and they bisect each other at right angles, forming two equal isosceles right triangles. If the side length is s, the area is s^2 and the perimeter is 4s; the diagonal length is s√2. The square is a special case of a rectangle and of a rhombus, and it has full symmetrical properties corresponding to the dihedral group D4.

In mathematics, a square refers to a perfect square, an integer that is the square of another integer. Examples: 0, 1, 4, 9, 16, 25. Square numbers appear in geometry and number theory; the square root of a square is an integer. The difference between consecutive squares is odd: (n+1)^2 − n^2 = 2n+1. Square numbers are closed under multiplication: the product of two squares is a square. They also arise in Pythagorean triples and in solving quadratic equations.

In urban design, a square is a public open space bounded by streets, used for assembly, commerce, or recreation. Squares often host monuments, fountains, or seating; they vary in scale and can be pedestrianized. Notable examples include Red Square in Moscow, Trafalgar Square in London, and Times Square in New York City. The term can also refer to central areas in cities that function as social or ceremonial centers, reflecting historical and cultural patterns of urban life.