Square

Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art.

[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:

Like all regular polygons, it is an isogonal figure, meaning that it has symmetries taking every vertex to every other vertex, and an isotoxal figure, meaning that it has symmetries taking every edge to every other edge.

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

[25] Graph paper, preprinted with a square tiling, has been widely used for data visualization using Cartesian coordinates[26] since its invention in 1794.

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

Ancient examples include the Egyptian pyramids,[32] Mesoamerican pyramids such as those at Teotihuacan,[33] the Chogha Zanbil ziggurat in what is now Iran,[34] 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,[35] the square bases of many Buddhist stupas,[36] and East Asian pagodas, buildings that symbolically face to the four points of the compass and reach to the heavens.

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

[39] 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.

[41][42] Artists whose works have used square frames and forms include Josef Albers,[43] Kazimir Malevich[44] and Piet Mondrian.

[45] Baseball diamonds[46] and modern boxing rings are square despite being named for other shapes.

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

[50] 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.

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

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

[55] Squares are a common element of graphic design, often used to give a sense of stability, symmetry, and order.

[68] For instance the illustration shows a diagonal square centered at the origin

][citation needed] The construction of a square with a given side, using a compass and straightedge, is given in Euclid's Elements.

Because a square is a convex regular polygon with four sides, its Schläfli symbol is {4}.

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

[88] In ancient Greek deductive geometry, 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.

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

[94] 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.

[103] Packing squares into other shapes can have high computational complexity: testing whether a given number of unit squares can fit into an orthogonally convex rectilinear polygon with half-integer vertex coordinates is NP-complete.

these numbers are:[112] Squares tilted at 45° to the coordinate axes are the metric balls for taxicab geometry, the

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

Among rectangles (top row), the square is the shape with equal sides (blue, middle). Among rhombuses (bottom row), the square is the shape with right angles (blue, middle).
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 .
A square with Gaussian integer vertices can be characterized by its center c and half-diagonal p . Either c and p will both be Gaussian integers, or (as in this case) both be Gaussian integers offset by ⁠ 1 / 2 ⁠ (1 + i ) .
The Calabi triangle and the three placements of its largest square. [ 84 ] 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. [ 100 ]
All 14 squares in a 3×3-square (4×4-vertex) grid
Three partitions of a square into similar rectangles
Points (red) at equal distance from a central point (blue) according to taxicab geometry