  We give a new existence proof for the rank 2^d even lattices usually called
the Barnes-Wall lattices, and establish new results on uniqueness, structure
and transitivity of the automorphism group on certain kinds of sublattices. Our
proofs are relatively free of calculations, matrix work and counting, due to
the uniqueness viewpoint. We deduce the labeling of coordinates on which
earlier constructions depend. Extending these ideas, we construct in dimensions
2^d, for d>>0, the Ypsilanti lattices, which are families of indecomposable
even unimodular lattices which resemble the Barnes-Wall lattices. The number
Upsilon (2^d) of isometry types here is large: log_2 (Upsilon (2^d)) has
dominant term at least (r/4) d 2^{2d}, for any r in [0, 1/2). The Ypsilanti
lattices may be the first explicitly given families whose sizes are
asymptotically comparable to the Siegel mass formula estimate (log_2(mass(n))
has dominant term (1/4) log_2(n) n^2). This work continues our general
uniqueness program for lattices, begun in Pieces of Eight. See also our new
uniquness proof for the E_8-lattice.
