  We connect the theories of the deformation of elastic surfaces and phase
surfaces arising in the description of almost periodic patterns. In particular,
we show parallels between asymptotic expansions for the energy of elastic
surfaces in powers of the thickness $h$ and the free energy for almost periodic
patterns expanded in powers of $\epsilon$, the inverse aspect ratio of the
pattern field. For sheets as well as patterns, the resulting energy can be
expressed in terms of natural geometric invariants, the first and second
fundamental forms of the elastic surface, respectively the phase surface. We
discuss various results for these energies and also address some of the
outstanding questions.
  We extend previous work on point (in 2D) and loop (in 3D) disclinations and
connect their topological indices with the condensation of Gaussian curvature
of the phase surface. Motivated by this connection with the charge and spin of
pattern quarks and leptons, we lay out an ambitious program to build a
multi-scale universe inspired by patterns in which the short (spatial and
temporal) scales are given by a nearly periodic microstructure and whose
macroscopic/slowly varying/averaged behaviors lead to a hierarchy of structures
and features on much longer scales including analogues to quarks and leptons,
dark matter, dark energy and inflationary cosmology. One of our new findings is
an interpretation of dark matter as the energy density in a pattern field. The
associated gravitational forces naturally result in galactic rotation curves
that are consistent with observations, while simultaneously avoiding some of
the small-scale difficulties of the standard $\Lambda$CDM paradigm in
cosmology.
