Plane (geometry)

Lore & Background

Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing notions such as similarity of shapes, the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are 'equal' (meaning congruent in shape and size, not just having the same area). Later, the plane was described in a Cartesian coordinate system, a system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines. The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat's analytic geometry was primarily two-dimensional and he did not publish the discovery. Both authors used a single (abscissa) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis. The concept of using a pair of fixed axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students.

Reader's Guide

The Euclidean plane is fundamental to geometry and mathematics, serving as the setting for classical Euclidean geometry and later analytic geometry. Its structure as an affine space with a metric allows the definition of circles, angles, and distances. The introduction of the Cartesian coordinate system enabled the algebraic treatment of geometric problems, linking algebra and geometry. The plane is also viewed as a field in the complex plane, where points can be multiplied and divided, leading to applications in complex analysis. In linear algebra, the plane is two-dimensional because every point can be described by a linear combination of two independent vectors. The dot product provides a way to compute lengths and angles. The plane contains infinitely many polytopes (polygons), including regular convex polygons, degenerate spherical polygons, and non-convex star polygons. The hypersphere in two dimensions is a circle, with area πr². Other curved shapes include conic sections such as ellipses, parabolas, and hyperbolas. The gradient in calculus is defined in rectangular coordinates, and line integrals and double integrals are computed over the plane.

Did You Know?

• A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.
• The set ℝ² of ordered pairs of real numbers, equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane.
• The complex plane is sometimes called the Argand plane, named after Jean-Robert Argand, though first described by Caspar Wessel.
• In two dimensions, there are infinitely many polytopes: the polygons, including regular star polygons with Schläfli symbols {n/m}.