In the history of science, culture, and arts, symmetries have been used as fundamental models of order. This raises the question of whether they were merely invented by humans to order the diversity of phenomena, whether they even arise only from an aesthetic need, or whether they are basic structures of nature that exist independently of humans. In antiquity, at any rate, knowledge, arts, and nature were understood from a common symmetrical basic order.

In this talk, the basic mathematical conceps of symmetry are first explained with illustrative examples up to the group-theoretical methods. On this basis, fundamental principles of physics such as the conservation laws, time, space, and charge invariance (CPT theorem), and space-time structures are introduced with symmetries. In the standard model of elementary particle physics, local gauge symmetries are used to characterise the fundamental forces in the quantum world. Symmetry breaking occurs, which is also used to explain the expansion of the universe. Molecular symmetry breaking (chirality) is at the beginning of biological evolution.

In addition to mathematical models, measurement and experiment have been decisive in deciding questions of symmetry in nature since Galileo's time. Today, the technical effort required for this is enormous and ranges from the laboratories of high-energy physics to observatories and satellites. Symmetry and symmetry breaking are proving to be complementary research principles for explaining the emergence of diversity and complexity in nature.

References: K. Mainzer, Symmetries in Nature (De Gruyter: New York 1996, German: De Gruyter: Berlin 1988); K. Mainzer, Symmetry and Complexity. The Spirit and Beauty of Nonlinear Science (World Scientific: Singapore 2005); K. Mainzer, Symmetry and complexity in dynamical systems, in: European Review (Academia Europaea) Vol. 13 Supplement 2 2005, pp. 29-48 (Cambridge University Press); M.C. Doncel, A. Herman, L. Michel, A. Pais (Eds.), Symmetries in Physics (1600-1980), Seminari d’Història de les Ciències Universita Autònoma de Barcelona (Bellaterra Barcelona Spain) 1987.