Elastic Sheets, Phase Surfaces, and Pattern Universes

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 ε, 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 two dimensional) and loop (in three dimensional) 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 multiscale 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 analogs 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 ΛCDM (cold dark matter) paradigm in cosmology.