The Euclidean plane needs two numbers per point
A Euclidean plane is Euclidean space in two dimensions, where two real numbers pin down every point. As an affine space it has parallel lines, and its distance function adds circles and angle measurement; the set of real number pairs with the dot product is the standard model every such plane matches.
Euclid's Elements covered flat geometry in Books I to IV and VI, treating similar shapes, parallels, the angle sum of a triangle, equal angles and areas, the three cases in which triangles count as equal, and, as Proposition 47, the Pythagorean theorem. Pick a Cartesian coordinate system and the plane becomes a Cartesian plane: each point gets signed distances to two perpendicular axes crossing at an origin, usually labelled x and y.
That idea appeared in 1637 in the work of Descartes and, independently, Pierre de Fermat, who also explored three dimensions but did not publish. Both actually used a single axis, measuring the second coordinate along lines not necessarily at right angles to it. The now-familiar pair of fixed axes came after Frans van Schooten and his students translated Descartes's La Géométrie into Latin in 1649. Polar coordinates offer another option, giving a point's distance from the origin and its angle from a reference ray.
Treat points as numbers that can be multiplied and divided and you get the complex plane, also called the Argand plane after Jean-Robert Argand, though the land surveyor Caspar Wessel described such diagrams first. They are often used to plot a function's poles and zeros. Linear algebra explains the count of two: any location can be written by mixing just two independent vectors, just as a rectangle's length is independent of its width.
The plane holds infinitely many regular polygons, plus star polygons with symbols like {5/2}. Its circle has circumference 2πr, and conic sections such as ellipses, parabolas and hyperbolas live there too. The dot product of two vectors equals the product of their lengths and the cosine of the angle between them, and a line integral through a gradient field depends only on the endpoints.
Source: Euclidean plane
