Mainzer, Klaus Symmetry and complexity in dynamical systems. (English) Zbl 1324.37002 Symmetry Cult. Sci. 26, No. 1, 5-38 (2015). Summary: Historically, static symmetric bodies and ornaments are geometric idealizations in Platonic tradition. Actually, symmetries are locally and globally broken by phase transitions of instability in dynamical systems generating a variety of new order and partial symmetries with increasing complexity. The states of complex dynamical systems can refer to, e.g., atomic clusters, crystals, biomolecules, organisms and brains, social and economic systems. The paper does not only discuss dynamical foundations of symmetry and complexity in nature. Simulations of symmetry and complexity in cellular automata are studied, in order to understand the algorithmic foundations of their emergence. Emergence of symmetry and complexity is an interdisciplinary challenge of nonlinear systems science. Philosophy of science analyzes the common methodological framework of symmetry and complexity. MSC: 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory 37N25 Dynamical systems in biology 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 35B36 Pattern formations in context of PDEs Keywords:local and global symmetries; symmetry breaking; complexity; pattern formation; local activity; cellular automata PDFBibTeX XMLCite \textit{K. Mainzer}, Symmetry Cult. Sci. 26, No. 1, 5--38 (2015; Zbl 1324.37002)