vierdimensions

Vierdimensionen is a term that refers to systems described by four independent coordinates. In mathematics this often means a four-dimensional Euclidean space or a general four-dimensional coordinate system with coordinates such as (x, y, z, w). In physics the phrase commonly denotes spacetime, which combines three spatial dimensions with one temporal dimension into a single four-dimensional manifold.

In Euclidean four-space, the distance between two points p = (x1, y1, z1, w1) and q = (x2, y2, z2, w2) is given by the square root of the sum of the squares of the coordinate differences: sqrt((dx)^2 + (dy)^2 + (dz)^2 + (dw)^2). This extends the familiar three-dimensional distance to an additional dimension and is used in geometry, data visualization, and related fields.

In physics, vierdimensionen frequently refers to spacetime. In special relativity, time is treated as a fourth dimension alongside space, with a metric that leads to the spacetime interval ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 (sign conventions vary). This framework preserves the invariant interval between events and unifies the laws of physics for all inertial observers. In general relativity, spacetime is a curved four-dimensional manifold described by a metric tensor g_{μν}, and gravity is interpreted as curvature of this four-dimensional fabric.

Historically, the concept emerged from attempts to unify space and time, notably by Hermann Minkowski in the early 20th century, and it remains foundational in modern physics. Four dimensions also appear in mathematics as the study of 4-manifolds and in theoretical models that extend beyond spacetime, such as higher-dimensional theories and certain geometric constructions.