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On the molecular level, for instance, the relatively-featureless isotropic-symmetry of liquid water may be broken by local translational pair-correlations (resulting in spherical reciprocal-lattice shells) as the liquid turns to polycrystal ice, and by both translational and rotational ordering (resulting in reciprocal-lattice spots) as the ice becomes a single crystal. A quasicrystal phase might have rotational without translational ordering, while a random-layer lattice might have rotational+translational ordering in one direction only. Thus even within a single layer of organization, broken symmetries play a role in the (at least temporary) development of order.

Hierarchical ordering in the layer just above a pair-correlated level may generally require a higher-level symmetry-break, which in turn gives rise to processes that select for inward-looking post-pair correlations as well as outward-looking pair-correlations on the next level up. Thus shared-electrons break the symmetry between in-molecule and extra-molecule interactions, bi-layer membranes allow symmetry between in-cell and out-cell chemistry to be broken, shared resources (like steady-state flows) may break the symmetry between in-tissue and external processes, metazoan skins allow symmetry between in-organism and out-organism processes to be broken, bias toward family breaks the symmetry between in-family and extra-family processes, membership-rules break the symmetry between in-culture and multi-cultural processes, etc.

1. ↑ P. W. Anderson (1972) "More is different", Science 177:4047, 393-396 pdf.
2. ↑ P. Fraundorf (2008) "The thermal roots of correlation-based complexity", Complexity 13:3, 16-26.
3. ↑ James P. Sethna (2006) Entropy, order parameters and complexity (Oxford U. Press, Oxford UK) (e-book pdf).
4. ↑ P. Fraundorf (2011/2012) "Metric-first & entropy-first surprises", arXiv:1106.4698 [physics.gen-ph].

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