Square

Square shapes are ubiquitous in the design of tiles, building floor plans, game boards, origami paper, quilting, etc.

[10] (The square root of 2, appearing in this formula, is irrational, meaning that it is not the ratio of any two integers.

[14] Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area.

[15] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:

is the distance from an arbitrary point in the plane to the i-th vertex of a square and

are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then

[27] Graph paper, preprinted with a square tiling, has been widely used for data visualization using Cartesian coordinates[28] since its invention in 1794.

[33] Many architectural structures from both ancient and modern cultures have featured a square floor plan, base, or footprint.

Ancient examples include the Egyptian pyramids,[34] Mesoamerican pyramids such as those at Teotihuacan,[35] the Chogha Zanbil ziggurat in what is now Iran,[36] the four-fold design of Persian walled gardens, said to model the four rivers of Paradise, and later structures inspired by their design such as the Taj Mahal in India,[37] the square bases of many Buddhist stupas,[38] and East Asian pagodas, buildings that symbolically face to the four points of the compass and reach to the heavens.

[40] In modern architecture, a majority of tall skyscrapers feature a square plan for pragmatic rather than aesthetic or symbolic reasons.

[41] On a smaller scale, the stylized nested squares of a Tibetan mandala, like the design of a stupa, function as a miniature model of the cosmos.

[43][44] Artists whose works have used square frames and forms include Josef Albers,[45] Kazimir Malevich[46] and Piet Mondrian.

[47] Baseball diamonds[48] and modern boxing rings are square despite being named for other shapes.

[50] The chessboard inherited its square shape from a pachisi-like Indian race game and in turn passed it on to checkers.

[52] The ancient Greek Ostomachion puzzle (according to some interpretations) involves rearranging the pieces of a square cut into smaller polygons, as does the Chinese tangram.

[53] Another set of puzzle pieces, the polyominos, are formed from squares glued edge-to-edge.

[54] Medieval and Renaissance horoscopes were often arranged in a square format, across Europe, the Middle East, and China.

For instance the illustration shows a diagonal square centered at the origin

More strongly, there exists a convex set on which no other regular polygon can be inscribed.

[67] In ancient Greek mathematics, the area of a planar shape was measured and compared by constructing a square with the same area by using only a finite number of steps with compass and straightedge, a process called quadrature or squaring.

Euclid's Elements shows how to do this for rectangles, parallelograms, triangles, and then more generally for simple polygons by breaking them into triangular pieces.

[68] Some shapes with curved sides could also be squared, such as the lune of Hippocrates[69] and the parabola.

In modern mathematics, this formulation of the theorem using areas of squares has been replaced by an algebraic formulation involving squaring numbers: the lengths of the sides and hypotenuse of the right triangle obey the equation

[73] Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully to square the circle, constructing a square with the same area as a given circle, again using finitely many steps with a compass and straightedge.

A construction for squaring the circle could be translated into a polynomial formula for π, which does not exist.

[79][80][81] Circle packing in a square is another packing problem with a similar flavor, where the goal is to fit as many unit circles as possible into a larger square of a given size, or again to minimize the uncovered area.

[82] Testing whether a given number of axis-aligned unit squares can be packed into an orthogonally convex rectilinear polygon with half-integer vertex coordinates is NP-complete.

[84] A relaxed version of the problem, called "Mrs. Perkins's quilt", asks for a subdivision of a square with integer side lengths into as few as possible smaller squares for which the greatest common divisor of the side lengths is 1.

[86] Squares tilted at 45° to the coordinate axes are the metric balls for taxicab geometry, the

[90] An octant is a regular spherical triangle consisting of three straight sides and three right angles.

The area of a square is the product of the lengths of its sides.
Classification of quadrilaterals by their symmetry subgroups. [ 18 ] The 8-fold symmetry of the square is labeled as r8, at the top of the image. The "gyrational square" below it corresponds to the subgroup of four orientation-preserving symmetries of a square, using rotations but not reflections.
The inscribed circle (orange) and circumscribed circle (pink) of a square (white)
Site of the Yongning Pagoda
plotted on Cartesian coordinates .
The Calabi triangle and the three placements of its largest square. The placement on the long side of the triangle is inscribed; the other two are not.
The Pythagorean theorem : the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse.
A circle and square with the same area
The smallest known square that can contain 11 unit squares has side length approximately 3.877084. [ 79 ]
Points (red) at equal distance from a central point (blue) according to taxicab geometry