In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Minkowski plane (named after Hermann Minkowski

Hermann Minkowski

Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

) is a two-dimensional affine space

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

 provided with a metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

 that is invariant

Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...

 under translations

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

.

Often one identifies the underlying affine space with the plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 R².
With this identification, it is easily seen that any Minkowski plane corresponds to a norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 on that plane, and vice versa.

A generalization of the Minkowski plane is the Benz plane

Benz plane

In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz. The term was applied to a group of objects that arise from a common axiomatization of certain structures and split into three families, which were introduced separately:...

.

External links

• On the Perimeter and Area of the Unit Disc, by J.C. Álvarez Pavia and A.C. Thompson

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