Introduction

In May 2016 an Academia Europaea symposium with the title ‘Symmetry, Proportion and Seriality: The Semantics of Mirroring and Repetition in Science and the Arts’ was organized at the Freiburg Institute for Advanced Studies (FRIAS). Its major aim was to bring together interdisciplinary perspectives on symmetry and related concepts, comparing humanities approaches with those current in the material sciences.

Symmetry is one of the key factors in a variety of scientific disciplines, social studies and humanities subjects. Mathematical models may exhibit symmetries; in architecture symmetry is a basic design feature; linguists discover iconic and symmetrical relationships in their objects of study; in chemistry and physics symmetrical and asymmetrical designs play an important role; in music and all the arts, symmetry is often considered the basis of aesthetic quality. Symmetry, and there are several types of symmetry that one might want to distinguish, can be set off against, but also paired with, two other features that play a comparable role in the sciences and the arts: proportion and seriality. Exact symmetry in some instances is too neat, too boring, or simply not possible; a set of proportional relationships, on the other hand, may be deemed crucial to a particular aesthetic effect, or for the categorization and understanding of hierarchical structures in mathematical idealizations, as well as in the natural world around us. Proportion can thus be regarded as a more general framework that allows one to set items in relationships to one another, with symmetry and mirroring being the most perfect of these relationships. The visual arts, especially film and dance, employ proportion and symmetry as kinetic rather than merely static modes. As regards seriality, the third concept in our title, it is a recursive application of symmetry and repetition, but also a type of design that operates dynamically rather than statically. Repetition bears an obvious relevance to the arts in experiments with seriality in (post)modernist painting, music and literature, but also in mathematics and physics, think of, for example, algorithmic structures or symbolic dynamics, and of time standards which rely on perfectly periodic motion on ever more minuscule atomic or nuclear scales.

Appropriately, this special issue starts with a poetic contribution by Mark McMorris, thus underlining the significance of symmetry and repetition for the arts. McMorrisʼs illustration, much in the vein of similar aesthetic strategies employed, for instance, by Minimal Artists such as Sol LeWitt, illustrates how complex the issues of repetition and seriality really are. On the one hand, and in a Platonic sense, each repetition conceives of its difference and uniqueness on the basis of a preconceived similarity and identity between a situation, an event, a motif and its ‘repetition’. On the other hand, however, as Gilles Deleuze (Reference Deleuze and Patton1994, 100ff.) has shown drawing on Nietzscheʼs rather than Platoʼs philosophy, both similarity and identity are thinkable only on the basis of a fundamental difference. Repetition, paradoxically, can only describe a unique series of events, and in a temporalized universe there cannot be any repetition in a strict sense. Hence, in philosophy, in the arts, and in the sciences temporality subverts the idea of repetition (and symmetry): ‘[A]ll things move and nothing remains still’, Heraclitus is quoted as stating by Socrates in Platoʼs Cratylus, thus emphasizing, almost proverbially, that ‘you cannot step twice into the same stream’ (Plato 1970, 402a). Repetition (and symmetry), therefore, only exist as cognitive products in acts of knowledge and (self-)reflexion. In this act of cognition two different and successive events, motifs, situations, etc., are perceived as situated in a temporal sequence, but, then, these two incidents are brought into such a relation with each other that it allows us to identify them as identical, serial, or symmetrical. By recognizing identities in repetition and sequences, so to speak, we assign a synthetic temporal relation to events, incidents and motifs that transcend phenomenological time and pure becoming. The psychological view brought forward, for instance, by Rosapia Lauro-Grottoʼs paper, demonstrates the crucial significance of symmetry in our cognition, including the workings of the now famous mirror neurons. In a similar vein, Fionn Murtaghʼs paper scrutinizes the theory of ‘bi-logic’ brought forward by Chilean psychoanalyst Ignacio Matte Blanco. Matte Blanco conceived of the symmetrical and the asymmetrical as the two modes of being, as he related symmetrical relations to the unconscious mind, while asymmetrical relations refer to logic and, thus, to the conscious sphere of human life.

Beyond that, cognitive predilection for proportion and symmetry appears perhaps optimally in architectural design and in music. One of the most significant insights to be garnered from the essays is that the term ‘symmetry’ has very different meanings both historically and across disciplines. As Hans Huber demonstrates, symmetry originally meant proportion and was a key factor in architectural aesthetics until it became focused on axial symmetry or mirroring. In tandem with this essay on architecture, Vasco Zara provides a context for the historical interplay of music theory and architecture in the Renaissance period.

In the natural sciences and in mathematics, the term expresses the invariancy of certain system properties under specific changes of the reference system (e.g. the symmetry of a sphere is formalized by its invariance under arbitrary rotations – a property a cube or a book do not have). This invariancy is, however, often anything but obvious, and needs to be discovered to formulate physical laws. Symmetries do not need to express global system properties, but can refer to specific instances of a physical systemʼs dynamics, and can be only approximately true and/or transient in nature. Yet even approximate symmetries may permit a faithful description of the system state on a certain coarse-grained level, in space and time. While repetition and seriality are concepts that approach symmetry, in abstract mathematical reference systems, temporality implies the bizarre absence of symmetry on the time axis, which we associate with the non-invertible arrow of time.

That invariance under substitution, i.e. symmetry, always hinges on the specific context defining ‘invariance’, and, in other words, is a matter of the chosen perspective, is Hartmann Römerʼs central observation. Unfolding a physicistʼs perspective upon symmetry, seriality (here considered as a special case of symmetry), and repetition, he reflects upon the subtle equilibrium between symmetries and their breakings to conceive our categories of beauty and of freedom.

Whether symmetries are pointing at real objects or rather at dynamically (in the sense of seriality, repetition and temporality) defined entities is the topic of Thierry Paulʼs contemplations in favour of a non-Platonist view on mathematical reality. Building on our modern understanding of nonlinear dynamical systems and of non-commutative structures in the quantum world he advocates the point of view that realism is intimately tied to movement.

This special issue also provides some extensive illustrations on the pervasiveness of symmetry and repetition in the arts. Longxi Zhangʼs essay on poetry demonstrates that parallelism (and antithesis) are key devices not only in Chinese poetry but also in neoclassical British poetry. The remaining two essays from the humanities deal with repetition and serialization. As Monika Schmitz-Emans demonstrates, experimental German poetry utilizes a number of strategies of reiteration and serialization. Angeli Janhsenʼs paper complements this diagnosis in reference to (post)modern art and provides a bracket for the discussion of symmetry, proportion, repetition and seriality in the arts. She links the technique of seriality with the history of painting, discussing the typical features of proportionality and symmetry foregrounded in the classical paintings of Claude Lorrain and how these patterns of symmetry are creatively modified, questioned, condensed and expanded in late twentieth-century and contemporary art.

In the humanities contributions, it emerges that symmetry, proportion, parallelism, repetition and seriality serve two main functions – first, aesthetic pleasure which, in accordance with many theories of aesthetics, is linked to balance and slight modification of the same; and, second, the suggestion of form-and-content correspondence. In fact, the whole issue of style in itself, from a linguistic perspective, can be considered a matter of repetition with variation. To the extent that discourses keep repeating well-known phrases, they are considered to be clichéd, boring and aesthetically defective; to the extent that metaphors are heaped on metaphors and the language bears little resemblance to standard usage, the text will be considered either as an example of enhanced rhetorical and artistic skill or as defective through an excess of innovation, especially if that deviation from ‘normal’ language is insufficiently motivated by content and generic context (more appropriate in the ode or the epic than in an essay or a novel). It would then also tend to be associated with the great and sublime. The golden mean of unassuming variation constitutes the aesthetically most pleasing stylistic expression. Repetition of linguistic material with slight variation is therefore a standard for artistic excellence in its own right.

The concepts of symmetry, proportion and repetition are not merely factors in artistic contexts and language as well as in cognition and psychology. While their importance has been exploited for the purposes of meaning and aesthetics in the arts and humanities, in the natural sciences the function of symmetry – perhaps in parallel to logics – relates to conceptual classification and the formulation of the fundamental laws of nature. The interplay of transient, approximate symmetries and their fading away or destruction is nowadays understood as the basic source of the diversity of patterns and forms in the natural world around us.