Euclidean plane

A Euclidean plane is a flat, two-dimensional geometric space. In this space, you only need two real numbers to find the exact position of any point. This space is considered an affine space, which means it includes the concept of parallel lines. It also possesses metrical properties. These properties are induced by a distance, allowing us to define angles and circles.

To navigate this plane, mathematicians often use a Cartesian coordinate system. This system uses two perpendicular lines called axes. These axes meet at a central point known as the origin, usually labeled (0, 0). Every point is identified by an ordered pair of real numbers. These numbers represent the signed distances from the point to the axes. You can also use a polar coordinate system. This system identifies a point using its distance from the origin and its angle relative to a reference ray.

Many different types of shapes can exist within a Euclidean plane. One group is called polygons, which are two-dimensional polytopes. Regular polygons have sides and angles that are all equal. Examples include the triangle, square, pentagon, and hexagon.

As you add more sides, you get shapes like the decagon or the icosagon. There are also non-convex regular polygons called star polygons. These shapes look like stars and use Schläfli symbols with rational numbers.

Beyond straight-sided polygons, the plane contains many curved shapes. A circle is a special shape in the plane. It is sometimes called a 1-sphere because it is a one-dimensional manifold. A circle has a length of 2πr and an interior area of πr².

Other curved shapes include conic sections. These include the ellipse, the parabola, and the hyperbola. These curves are fundamental to understanding how shapes move and change in two-dimensional space.

The study of the plane has a long history. Euclid explored two-dimensional geometry in his work, *Elements*. He developed ideas about the Pythagorean theorem, similarity, and the sum of angles in a triangle. Much later, in 1637, René Descartes and Pierre de Fermat independently developed the coordinate system. Descartes published his ideas in *La Géométrie*. In 1649, Frans van Schooten translated this work into Latin. This translation helped clarify the use of two fixed axes.

Another important version of the plane is the complex plane. In this view, the plane is treated as a field. This allows any two points to be multiplied or divided. This is often called the Argand plane because of Argand diagrams. These diagrams were named after Jean-Robert Argand, who published his work in 1806. However, the Danish-Norwegian mathematician Caspar Wessel described them earlier in 1797. These diagrams are used to plot the positions of zeroes and poles of a function.

In the field of linear algebra, the plane is viewed through the lens of independence. The plane is two-dimensional because the length of a rectangle is independent of its width. Technically, every point can be described as a linear combination of two independent vectors. You can visualize a vector as an arrow with a specific direction and magnitude. The dot product is a way to relate these vectors. It helps define the angle between them and the Euclidean length of the vector.

The Euclidean plane also connects to advanced calculus and topology. In calculus, you can use line integrals and double integrals to study fields within the plane. Green's theorem relates a line integral around a closed curve to a double integral over the region it encloses. In topology, the plane is a unique contractible 2-manifold. It has a special property: if you remove a single point, the remaining space stays connected, but it is no longer simply connected.