fourdimensional

Fourdimensional refers to something characterized by four dimensions. In mathematics, the primary context is Euclidean four-space, R4, consisting of all ordered quadruples (x1, x2, x3, x4). Distinguishing between four spatial dimensions and spacetime is common in physics: in relativity, time is treated as the fourth dimension combined with the three spatial dimensions to form four-dimensional spacetime.

In geometry, four-dimensional space extends the concepts of points, lines, planes, and solids to four coordinates. Classic objects include the four-dimensional analog of a cube, called a tesseract or hypercube, which has 16 vertices and 8 cubical cells. Cross-sections and projections into 3D help visualize 4D objects. Rotations in four dimensions form the group SO(4). Quaternions provide a useful 4D algebra that models three-dimensional rotations; they are a key example of a four-dimensional number system.

Four-dimensional manifolds generalize curves and surfaces to higher dimensions and are central in topology and geometry. In physics, Minkowski spacetime is a four-dimensional model combining space and time with a specific metric signature; general relativity treats spacetime as a dynamic four-dimensional manifold whose curvature encodes gravity. In high-energy and theoretical physics, additional dimensions beyond four are proposed in various theories, but four-dimensional models are foundational in many areas.

Historically, the study of four-dimensional geometry emerged in the 19th century, with contributions from mathematicians such as Hamilton and Schläfli; popularization occurred through works like Flatland and Henry Sidgwick?—though common references point to Early popularizations by writers like Edwin A. Abbott and later Auguste?—the exact names vary. Today, four-dimensional concepts are standard in mathematics and physics, with practical visualizations relying on projections, slices, and color coding to represent the fourth dimension.