Some rectangles cross themselves, and some have no right angles at all
Everyone knows a rectangle has four right angles. Geometers are less sure. A crossed rectangle folds over itself like a bow tie, and on curved surfaces the shapes called rectangles have equal angles that are not right angles. One definition even replaces angles entirely with lines of symmetry.
In ordinary flat geometry a rectangle is a quadrilateral with four right angles. Equivalent definitions call it an equiangular quadrilateral, since four equal angles must each be 90 degrees, or a parallelogram with a single right angle. When all four sides match it becomes a square, and a rectangle that is not square is sometimes called an oblong. The name comes from the Latin rectangulus, joining rectus, meaning right or proper, with angulus, angle.
There are several neat tests. A convex quadrilateral is a rectangle exactly when it is a parallelogram with diagonals of equal length, or a quadrilateral whose diagonals are equal and cut each other in half, among other criteria. Every rectangle is cyclic, meaning all four corners sit on one circle, and it has two mirror lines plus a half-turn rotational symmetry. It is also a special parallelogram, which is itself a special trapezium, a shape North Americans call a trapezoid.
Stretch the rules and odd cousins appear. A crossed rectangle uses two opposite sides of a rectangle together with its two diagonals, producing a self-intersecting figure whose angles are not right angles, though opposite ones match. In spherical, elliptic and hyperbolic geometry, the so-called rectangles keep equal opposite sides and equal angles but lose their right angles. Michael de Villiers proposed defining a rectangle as any quadrilateral with a symmetry axis through each pair of opposite sides, a definition that covers both the familiar and the crossed kinds.
Rectangles also pair off with rhombi. The dual polygon of a rectangle is a rhombus, and joining the midpoints of a rectangle's sides in order produces a rhombus, while doing the same to a rhombus yields a rectangle. They also feature in many tiling puzzles, from covering the plane with rectangles to filling a rectangle with other polygons.
Source: Rectangle
