  A lattice L is spatial if every element of L is a join of completely
join-irreducible elements of L (points), and strongly spatial if it is spatial
and the minimal coverings of completely join-irreducible elements are
well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular
lattice can be embedded, within its variety, into an algebraic and spatial
lattice. We extend this result to n-distributive lattices, for fixed n. We
deduce that the variety of all n-distributive lattices is generated by its
finite members, thus it has a decidable word problem. This solves two problems
stated by Huhn in 1985. We prove that every modular (resp., n-distributive)
lattice embeds within its variety into some strongly spatial lattice. Every
lattice which is either algebraic modular spatial or bi-algebraic is strongly
spatial. We also construct a lattice that cannot be embedded, within its
variety, into any algebraic and spatial lattice. This lattice has a least and a
largest element, and it generates a locally finite, join-semidistributive
variety.
