  We present a complete classification of the geometry of the mutually
complementary sets of entangled and separable states in three-dimensional
Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis
begins by finding the geometric structure of the pure product states in a given
three-dimensional Hilbert subspace, which determines all the possible separable
and entangled mixed states over the same subspace. In bipartite systems, we
characterise the 14 possible qualitatively different geometric shapes for the
set of separable states in any three-dimensional Hilbert subspace (5 classes
which also appear in two-dimensional subspaces and were found and analysed by
Boyer, Liss and Mor [Phys. Rev. A 95:032308, 2017], and 9 novel classes which
appear only in three-dimensional subspaces), describe their geometries, and
provide figures illustrating them. We also generalise these results to
characterise the sets of fully separable states (and hence the complementary
sets of somewhat entangled states) in three-dimensional subspaces of
multipartite systems. Our results show which geometrical forms quantum
entanglement can and cannot take in low-dimensional subspaces.
