4space

4space commonly refers to four-dimensional Euclidean space, denoted R^4, which consists of all ordered quadruples of real numbers. It is the natural extension of three-dimensional space used in geometry, physics, and computer graphics to include a fourth dimension. The standard distance between two points a = (a1, a2, a3, a4) and b = (b1, b2, b3, b4) is the Euclidean norm sqrt((a1−b1)^2 + (a2−b2)^2 + (a3−b3)^2 + (a4−b4)^2). The inner product ⟨a, b⟩ = a1b1 + a2b2 + a3b3 + a4b4 induces the norm ||a|| = sqrt(⟨a, a⟩). Geometric objects such as lines, planes, and hyperplanes are defined by linear equations in four coordinates; four-dimensional analogues of polygons include 4D polytopes, such as the tesseract.

In mathematics, 4space is studied within higher-dimensional geometry and topology, with topics including projection, stereographic mapping, and the behavior of shapes under rotations and reflections. The group of isometries of R^4 combines translations with the orthogonal group O(4), and subgroups preserve particular structures or orientations. In applications, four-dimensional representations appear in data analysis, physics, and computer graphics, where four coordinates may encode time alongside three spatial dimensions or serve as an abstract feature space that is reduced for visualization.

Outside of mathematics, 4space is also used as a branding or project name by various organizations and products, chosen to evoke ideas of space, openness, and modern design. The usage is not universally tied to the mathematical concept. See also higher-dimensional space, four-dimensional geometry, and the tesseract.