A square is the only regular polygon with all right angles equal
Four equal sides and four right angles make a square both rectangle and rhombus. Its interior, central, and exterior angles are all ninety degrees—the only regular polygon where those three match. Area is side length times itself, which is why algebra calls that operation squaring.
Many equivalent tests confirm the shape: a rhombus with a right angle, a rectangle with equal sides, or a quadrilateral whose equal diagonals perpendicularly bisect each other. Diagonals also bisect the corner angles into forty-five-degree pairs, and opposite sides stay parallel. All squares are similar; one length fixes size. Side times root two gives the diagonal, and that root two irrationality was already approximated in Babylonian mathematics. A four-by-four square is equable—area equals perimeter—sharing that integer-rectangle trait only with a three-by-six.
Among quadrilaterals the square is most symmetrical: eight rigid motions map it to itself, forming the dihedral group of order eight. Reflections and ninety-degree rotations permute vertices and edges transitively. Wallpaper groups p4, p4m, and p4g must use a square unit cell. Affine maps can send a square to any parallelogram; projective maps can send it to any convex quadrilateral, so perspective can disguise one as the other. Taxicab and Chebyshev metrics even treat squares as their balls.
Equal squares tile the plane and dominate graph paper, pixels, boards, and floors. The area formula powered classical attempts to square the circle with compass and straightedge—now known to be impossible—and still frames problems about packing unequal squares into a square. A square maximises area for a given perimeter among quadrilaterals and minimises perimeter for a given area, saturating the isoperimetric inequality exactly when the figure is square. Simple on a screen, it is geometry's tightest four-sided compromise.
Source: Square
