The meaningfamily of symmetry was developed through the ages in an interestic way. The main steps were made in both science and art. Some meanings also appeared “between”: these are associated with both sides of culture.
MATH/SCIENCE

between

ART


GREEKS

common measure

proportion


VITUVIUS

proportion
(theory)


RENAISSANCE

proportion


MODERN


– geometry

mirror sym.
rotosymm.

balance
cyclicity

proportion,
harmony

– crystallogr.

periodic sym.

rhythm
repetition


– physics

invariance

archetype

(Based on a slide that I presented at the first congress of ISIS,Budapest , 1989)
 Perhaps surprisingly to many people, the classical Greek term symmetria (summetria) did not refer to “bilateral symmetry” at all. Syn + metron is simply “common measure”. This concept, together with asymmetria, became important in geometry. following the discovery that two pairs of lengths could be both:
 – commensurable, e.g., 1 unitlength and 2 unitlength
 – incommensurable, e.g, the side and the diagonal of a square, i.e. 1 and Root(2).
 We may believe that the proof of the latter led to modern mathematics (sic!). The fact that the ratio of the side and the diagonal of a square (or of a regular pentagon) cannot be expressed as a/b, where a and b are integers was made by the method of indirect proof. First it was supposed that, to the contrary, there exist such a and b, but it led, after some logical steps, to contradiction. This fact meant that there are no such a and b. During this proof, logical operations were made with nonexisting objects. Thus, this “new geometry” is not any more a practical field (geo + measure), but abstract science with a deductive system.
 The expected good case was symmetria (commensurability), while the discovery of the new possibility of asymmetria (incommensurability) made some confusions. First of all, it “disturbed” the view of the Pythagoreans that everything can be expressed as ratios of integers, from musical harmonies to the motion of stars. This led very quickly to an aesthetical meaning of the same term:
 – symmetria refers to good ratios,
 – asymmetria refers to bad ratios.
 See the related algebraic terminology: rational vs. irrational (sic!) numbers.
 The early works of the Pythagoreans (6th c. B.C.) do not survive. Thus, we should consider later sources. It is important that in the 5th and 4th cc., Plato and Aristotle used the term symmetria in both senses: as a geometrical term and as an aesthetical concept. In fact, Aristotle claimed that there are three species of beauty:
 – taxis (order – it is closer to the modern concept of symmetry than symmetria)
 – symmetria (proportion)
 – horismenon (definiteness).
 In Latin the term symmetria had much less importance since different expressions were used:
 – commensuratio (the geometrical meaning of symmeria)
 – proportio (the aesthetical meaning of symmetria).
 My view if that the term symmetria survived because of Vitruvius‘ terminology (De architectura libri decem, 1st c.). He used both symmetria and proportio, but in slightly different senses:
 – symmetria refers to the theoretical aspects of proportions.
 – proportio refers to the practical aspects.
 I located many usages of the derivatives symmetria that are earlier than those ones which are given in major etymological dictionaries as the first usages outside Greek and Latin. All of these expressions are related to the modern translations of the Vitruvian text. In fact, the translators did not introduce the deritaves of porportion in the case of symmetria, since Vitruvius made distinction between the two concepts.
 However, this distinction was not relavent in later ages. The term symmetria became an “empty niche” for later usages … and mathematician adopted it. A mirror symmetric (bilaterialsymmetric) object has two equal halves and such a half can be considered as the unit of “common measure” for the entire object. In the case of nfold symmetry, we have n equal parts and one of them can be considered as unit of the “common measure”. Geometrical and crystallographic motivation helped the introduction of “crystallographic symmetries”, including translatary symmetries (periodic structures). In all above cases, symmetry can be interpreted as a transformation that moves the figure into itself, i.e., it reamains unchanged or, using another word, invariant.
 The next generalization of the concept of symmetry was made in physics where not only the invariance of geometrical shape, but also physical proporties were considered. In fact invariances and conservation laws play a major role in physics, According to E. P. Wigner , who earned the Nobelprize for his symmetryrelated works, there are three important levels:
 – events
 – physical laws
 – symmetries, the laws of laws.
 Symmetry, having roots in both art and science, may help the intersicsiplinary cooperation. In fact, symmetry may serve as a “bridge” between the two “hemispheres” of culture.
 Let it be … symmetry…
 References:
 Nagy, D. (1995) The 2500year old term symmetry in science and art and its `missing link’ between the antiquity and the modern age, Symmetry: Culture and Science, 6, No. 1, 1828.
 Nagy, D. (1996) QuasiSymmetrien und dynamische Symmetrien: Über die Entwicklung der Symmetrie in der Natur und deren Wiederspigelung in menschlichen Begriffen, [Quasisymmetries and dynamic symmetries: On the development on symmetry in nature and its reflection in human concepts, in German], In: Hahn, W., and Weibel, P., eds., Evolutionäre Symmetrietheorie: Selborganisation und dynamische Systeme, Stuttgart: Hirzel, 219228
 Nagy, D. (1996) The Western symmetry and the Japanese katachi shake hands: Interdisciplinary study of symmetry and morphological science (formology), In: Ogawa, T., Miura, K., Masunari, T., Nagy, D., eds., Katachi U Symmetry, Tokyo: Springer, 2746.
 Nagy, D. (1997) Symmetry: From the birth of Greek mathematics and aesthetics to a modern bridge between science and art, In: Koptsik, V. and Riznichenko, G., eds., Mathematics and Art: International Conference, Proceedings, Moscow: Moscow State University, 6469.
 Nagy, D. (1998) (Dis)symmetry: Mathematics and design, Euclidean vs. Vitruvian mathematics, In: Barallo, J., ed., Mathematics and Design 98: Proceedings of the Second International Conference, [June 14, 1998], San Sebastian: Universidad del Pais Vasco, 1725.
 Nagy, D. (2002) Architecture, mathematics, and a “symmetric link” between them (From the Atomium building to the MatOmium project), Symmetry: Art and Science, 2002, Nos. 14, 3162 [ = MatOmium EuroWorkshop, Brussels, April 913, 2002; Opening plenary talk of the workshop].
 Nagy, D. (2007) Forma, harmonia, and symmetria (With an appendix on sectio aurea), Symmetry: Art and Science, 2007, Nos. 24, 1941 [Plenary talk at the Seventh Interdisciplinary Symmetry Congress and Exhibition, Buenos Aires, Argentina, November 1117, 2007].