There are no finite-faceted regular tessellations of hyperbolic space of dimension 5 or higher.

In general the orbits are unions of Hermitian symmetric spaces of lower dimension.

How this generalizes in Euclidean spaces of higher dimension is illustrated in Fig. 5.2.

Let be a projective space of dimension .

For finite projective spaces of dimension over a field we have:

An affine space of positive dimension is not complete.

More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension.

The Poisson summation formula holds in Euclidean space of arbitrary dimension.

This description generalizes to complex projective space of higher dimension.

Especially in spaces of higher dimension, elliptic geometry is called projective geometry.