Carl Friedrich von Weizsäcker Zentrum, Universität Tübingen, Doblerstraße 33, 72074 Tübingen;
TUM Senior Excellence Faculty, Arcisstr. 21, 80333 München; European Academy of Sciences and Arts,
St.Peter Bezirk 10, 5020 Salzburg

In this review article, symmetry and symmetry breaking are considered as complementary principles in science and arts. It starts with symmetry and symmetry breaking in early world views of nature and art. Then, symmetries are definied as fundamental structures of mathematics. Mathematical models of symmetry and symmetry breaking are used to explain the emergence of space-time and matter in modern physics. Even molecular structures in chemistry are distinguished by mathematical symmetries. Their elegance and beauty seem to realize aesthetical categories in nature. In biological evolution, the question arises how symmetry breaking (e.g. molcular chirality) can be explained. In modern arts, symmetry and symmetry breaking are „hidden“ structures which can be found in music, painting, and architecture. In a philosophical outlook, symmetry and symmetry breaking are highlighted as regulative guiding ideas of research.

Platonic bodies, harmony, symmetry groups, automorphism, space-time symmetries, gauge symmetries, cosmological symmetry principle, global and local symmetries, spontaneous symmetry breaking, chirality.

Symmetries are used in the history of science and culture 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, art and nature were understood from a common symmetrical basic order. In modern times, this unity of natural and human sciences breaks down. In art, symmetries and symmetry calculations are related to subjective judgements of taste. In mathematics and the natural sciences, symmetries and symmetry breaking remain fundamental principles of describing nature, the application of which ranges from the formation of primordial matter to the evolution of life. In fact, current discoveries and laws in cosmology, physics, chemistry and biology are related to symmetry and symmetry breaking. It is symmetry breaking, according to the thesis in many of the author's books, that gives rise to diversity, complexity and new structures in nature - from physics and chemistry to biology and brain research. Without symmetry breaking, the world would remain invariant and unchanged.  Mathematical structures make these interdisciplinary connections in nature and art transparent. (Johnson et al., 2022; Conway et al., 2008; Ball 2016).

Audretsch, Jürgen/Mainzer, Klaus (Eds.)(1994), Philosophie und Physik der Raum-Zeit. Grundlagen der exakten Naturwissenschaften Bd. 7, Mannheim: B.I. Wissenschaftsverlag 2nd ed.
Audretsch, Jürgen/ Mainzer, Klaus (Eds.) (1990), Vom Anfang der Welt. Wissenschaft, Philosophie, Religion, Mythos, München: C.H. Beck 2nd ed.
Audretsch, Jürgen/Mainzer, Klaus (1996), Wieviele Leben hat Schrödingers Katze? Zur Physik und Philosophie der Quantenmechanik, Heidelberg: Spektrum Akademischer Verlag 2nd ed.
Ball.P (2016) Patterns in nature, University of Chicago Press, US.
Bernstein, Joseph (1974), Spontaneous symmetry breaking, gauge theories, the Higgs mechanism
and all that, in: Revise Reports of Modern Physics 46, 7-48.
Claus, R. (1980), Symmetrie in der Musik. Zur Anwendung gruppentheoretischer Methoden, in: A. Preisinger(ed.),Symmetrie, Springer: Vienna/New York.
Conway, J.H., Heidi Burgiel , Chaim Goodman-Strauss (2008) The symmetries of things, Taylor & Francis Inc.
Commins, E.D.; Bucksbaum, P.H. (1983), Weak Interactions of Leptons and Quarks, Cambridge University Press: Cambridge.
Doncel, Manuel G.; Hermann, Achim:  Michel, Louis; Pais, Abraham (Eds.) (1987), Symmetries in Physics 1600-1980, Bellaterra (Barcelona).
Ehlers, Jürgen (1973), The Nature and Structure of Spacetime, in: Mehra, Jagdish (Ed.), The Physicist’s Conception of Nature, Dordrecht: Kluwer Academic Publisher.
Eudoxos (1966), Die Fragmente des Eudoxos von Knidos (Ed. transl. comm. François Lasserre), De Gruyter: Berlin.
Feynman, Richard Philips; Leighton, Robert; Sands, Matthew (1966), The Feynman Lectures on Physics, Reading, Mass. [et al.], Addison-Wesley.2nd ed.
Gell-Mann, Murry; Ne’eman, Yuval (1964), The Eightfold Way, New York.
Georgi, Howard; Glashow, Sheldon Lee (1974), Unity of all elementary-particle forces, in: Phys. Rev. Lett. 32, 438-441.
Georgi, Howard (1980), Why unify? In: Nature 288, 649-651.
Greenberg, O.W. (2002): CPT violation implies violation of Lorentz invariance. in: Physical Review Letters  89,  2002, 231602.
Groom, D.E. et al. (2000), Review of Particle Physics, in: European Physics Journal C15, 1-4.
Heisenberg, Werner (1959), Wandlungen in den Grundlagen der Naturwissenschaften, Stuttgart: S. Hirzel: 9th ed.
Higgs, Peter Ware (1964), Broken symmetries, massless particles and gauge fields, in: Phys. Lett. 12, 132.
Hollas, John Michael (1975), Die Symmetrie von Molekülen, Berlin, De Gruyter.
Itzyson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory, New York: McGraw-Hill.
Jencks, Charles (1978), Die Sprache der postmodernen Architektur. Die Entstehung einer alternativen Tradition, Stuttgart: DVA.
Johnston, I.G., Kamaludin Dingle, Sam F. Greenbury and Ard A. Louis (2022) Symmetry and simplicity spontaneously emerge from the algorithmic nature of evolution, PNAS, 119 (11) e2113883119, https://doi.org/10.1073/pnas.2113883119
Klee, Paul (1928), Exakte Versuche im Bereich der Kunst, in: Bauhaus. Zeitschrift für Bau und Gestaltung 2 No. 2/3, Dessau.
Le Corbusier (1925), Urbanism, in: U. Conrads (ed.), Programme und Manifeste zur Architektur des 20. Jahrhunderts, Bauwelt Fundamente vol. 1, Berlin: Ullstein 1964.
D. Lee, T.D.; C. N. Yang, C.N. (1956), Question of Parity Conservation in Weak Interactions, in: Phys. Rev. 104, 254; Erratum Phys. Rev. 106, 1371 (1957).
Mainzer, Klaus (1980), Geschichte der Geometrie, Mannheim: B.I. Wissenschaftsverlag.
Mainzer, Klaus (1981), Grundlagenprobleme in der Geschichte der Exakten Wissenschaften, Konstanz: Universitätsverlag.
Mainzer, Klaus (German: 1988, English: 1996), Symmetries of Nature, Berlin/New York: De Gruyter:  Berlin.
Mainzer, Klaus; Hill, Craig; Müller, Achim (2013), Challenges of complexity in chemistry and beyond, in: Hill, Craig; Musaev, Djamaladdin (eds.), Complexity in Chemistry and Beyond: Interplay Theory and Experiment, New York: Springer, 1-28. Fig. 6 copyright Achim Müller (University of Bielefeld).
Mainzer, Klaus (2014), Die Berechnung der Welt. Von der Weltformel zu Big Data, München: C.H. Beck.
Mainzer, Klaus (2005), Symmetry and Complexity. The Spirit and Beauty of Nonlinear Science, World Scientific Publishing Singapore.
Mittelstraß, Jürgen (1970), Die Rettung der Phänomene. Ursprung und Geschichte eines antiken Forschungsprinzips, Berlin: De Gruyter.
Noether, Emmy (1918), Invariante Variationsprobleme, in: Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse, 235-257.
Perkins, Donald H. (2000), Introduction to High Energy Physics, Cambridge University Press: Cambridge 4th edition.
Primas, Hans (1985), Kann Chemie auf Physik reduziert werden? Erster Teil. Das Molekulare Programm, in: Chemie in unserer Zeit 19 4, 100-119.
Quack, Martin (1986), On the measurement of parity violating energy difference between enantiomers, in: Chemical Physics Letters 132  2, 147-153.
Schmutzer, Ernst (1972), Symmetrien und Erhaltungssätze der Physik, Pergamon Press: Oxford, Vieweg: Braunschweig.
Schrödinger, Erwin (1962), Was ist ein Naturgesetz? München: R. Oldenbourg.
Tranter, G.E.  (1986), Paritätsverletzung: Ursache der biomolekularen Chiralität, in: repr. Chem. Techn. Lab. 34 no. 9, 866.
van Doesburg, Theo; van Eesteren, Cornelis (1924), Auf dem Weg zu einer kollektiven Konstruktion, in: De Stijl 12, Issue 6/7.
Warwick, Roger; Williams, Peter (Ed.) (1973), Gray’s Anatomy, Edinburgh: Longman.
Weinberg, Steven (1985), Vereinheitlichte Theorie der elektro-schwachen Wechselwirkung, in: Dosch, Hans Günter (Ed.), Teilchen, Felder und Symmetrien. Spektrum der Wissenschaft, Heidelberg 2nd ed.
Werker, Wilhelm (1922), Studien über Symmetrie im Bau der Fugen und die motivische Zusammengehörigkeit der Präludien und Fugen des Wohltemperierten Klaviers von J.S. Bach, Leipzig: Breitkopf und Härtel.
Weyl, Hermann (1918), Gravitation und Elektrizität, in: Sitz. Ber. d. Preuß. Akad. d. Wissensch., 465-480.
Weyl, Hermann (2015), Symmetry, Berlin: De Gruyter.

Copyright: © 2022. This is an open-access article distributed under the terms of the Creative Commons Attribution License. (https://creativecommons.org/licenses/by/4.0/).