1 Representations of 4-D Spaces

This is a contributed topic entry on representing four-dimensional and higher-dimensional space structures.

1.1 2D and 3D Representations of 4-D and Higher-Dimensional Space Structures

The “representation” of 4-D and higher-dimensional space structures is a subject of interest to mathematical physicists, mathematicians, and artists. See, for example, artistic and architectural discussions of higher-dimensional representations.

A somewhat artistic rendering and animation of such a representation of a four-dimensional “octacube” by a mathematical physicist, together with a static representation by a mathematician, is presented in a related exposition.

1.2 Stereographic Projection

Intuitively, stereographic projection is a mapping that projects a sphere onto a plane. It is widely used in complex analysis, cartography, geology, and other fields. In practice, such projections may be produced computationally or constructed graphically using a stereonet. A famous historical illustration is the one by Rubens for Opticorum libri sex philosophis juxta ac mathematicis utiles, by François d’Aiguillon.

The animation link in the previous subsection uses Ocneanu’s method of windowed, radial stereographic projection. This projection method has been proposed as a useful way to represent four-dimensional solids because it can display two-dimensional walls of three-dimensional cells rather than only a one-dimensional edge framework.

Other projection methods may also be used to visualize higher-dimensional Euclidean or Riemannian structures. Such questions arise naturally when one attempts to visualize geometric structures used in mathematical physics, including Dirac particles in Riemannian space-times.

In four dimensions, a metric tensor may be represented in coordinates by a symmetric 4 × 4 matrix,

Because the metric tensor is symmetric, it has ten independent components in four dimensions. The metric components determine lengths and angles locally; curvature, however, is determined by derivatives of the metric through the connection and curvature tensors, not simply by the magnitudes of the metric components themselves.

The same idea extends to an N-dimensional manifold, where a symmetric metric tensor has

independent components.

The Riemannian exponential map, together with its local inverse, the Riemannian logarithm map, can be used in methods for visualizing metric tensor fields. One example is a metric-sphere glyph, in which the tensor field is interpreted through the local metric geometry of the manifold.

William Kingdon Clifford translated and discussed Riemann’s geometrical ideas in the nineteenth century and developed speculative physical interpretations in which matter and physical interactions were associated with curvature of space. These ideas anticipated later geometric approaches to gravitation and higher-dimensional theories.

For infinite-dimensional Riemannian manifolds, visualization is necessarily more indirect, since there is no literal finite-dimensional spatial representation of the full geometry.