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# Symmetry and beauty – A bridge between science and art?

A concept that moves scientists and artists alike and which seems to be suitable like no other to strike a rare discursive bridge between their two disciplines is the concept of beauty, in a narrower context “symmetry”. Most of the ancient and modern art conceptions regard symmetry as an essential criterion of beauty. Theoretical physics […]

A concept that moves scientists and artists alike and which seems to be suitable like no other to strike a rare discursive bridge between their two disciplines is the concept of beauty, in a narrower context “symmetry”. Most of the ancient and modern art conceptions regard symmetry as an essential criterion of beauty. Theoretical physics of the twentieth century, on the other hand, discovered in it a grounding principle.

The term is derived from the ancient Greek *symmetría*, a combination of *syn *(together) and *métron *(the right measure). Symmetry thus means “plane” or “uniform measure”. In the ancient concept of art symmetry described the ideal proportions of ratios of length and distances in sculptures, pictures, or buildings. Just like the perfect harmonies in music these find their expression in corresponding numeric ratios. The lead model thereby constituted the human body. For example, the ratio of the length of the arm to the whole body is a quarter. And when the arms and legs are outstretched, their ends describe exactly a square or alternatively a circle with the navel as its center. It was not by chance that the Renaissance scholar which is still regarded as one of the greatest artist and equally scholar in history emphasized this realization: Leonardo da Vinci. Already in antiquity two further concepts for symmetry supplemented the one of ideal proportions: reflection symmetry, as expressed in the relationship of left and right body parts, and the balance of opposites that can be found, for example, in Greek medicine and its teachings on body fluids.

The essential endeavor of scientists is to work out simple structures and processes within the confusing complexity of natural phenomena. Thus most theoretical physicists have the deep faith that nature, in spite of the multiplicity of its phenomena, is simple in its fundamental structure. And in this simplicity which finds its correspondence in the mathematical structures of physical theories they aim to discover the true beauty of nature. It is ironic that beauty express itself precisely through the discipline, whose grammar many artists have never appreciated or mastered. “Simple” for a physicist means just about anything that can be represented mathematically in exactness. Not that is simple, which nature presents to us directly, but the scientists have to first separate the colorful mixture of the phenomena, i.e. free the important from all unnecessary accessories (such as friction in free fall), until the “simple” processes reveal themselves. Only the simple can then appear as “beautiful”.

The separation of the “unnecessary accessory” is most easily performed in astronomy (there is no friction in outer space), which hence constituted the starting point of the scientific revolution. Johannes Kepler was so enthusiastic about the beauty and simplicity of the heavenly motions that he considered his laws of planetary movement the expression of supreme divine principles. And Newton not only provided physicists the mathematics which enable them to calculate the planetary motion, but also gave legitimacy to Galilei’s rather daring view that “the book of nature is written in the language of mathematics”. He thus also justified science’s claim of being able to derive and calculate all natural phenomena, a claim which it still maintains in one form or another today and which made it the most influential social and intellectual force of modernity. No less was Einstein moved by the stringency and sublimity expressed in the mathematical expressions of natural laws. His general theory of relativity, which gave the phenomenon of gravitation a wonderful geometric formulation, is still regarded as one of the most exalted and beautiful theories of nature. Heisenberg contended along these lines: “The final theory of matter will be characterized by a series of important symmetrical orders, similar to what Plato wrote.” But these symmetries are not necessarily visually perceptible anymore, as he goes on to explain: “These symmetries cannot any longer by described by figures and images, as it was the case for the Platonic bodies, but by equations.“

As Heisenberg implies, the concept of symmetry in the natural sciences is different from that in art. It is less about proportions or balance but about order and structure. In the mathematical assessment of nature symmetry describes an important aspect in the characterization of the structure and dynamics of natural objects. An example is the classification of crystals, as it was performed in the 18th century and 19th century. The symmetries of crystals express themselves by the fact that rotations around certain angles (and axes) do not alter their appearance. As Kepler already discovered, snow crystals, with all their individuality, are always symmetrical like a hexagon (the reason being the particular form of the water molecule). The influence of crystallography was also clearly evident when a generalized mathematical concept of symmetry was developed in the nineteenth century: invariance against transformations. A structure is considered symmetrical when certain rotations transform it back into its original form.

Just like the rotations of bodies, algebraic equations, differential equations, and general geometric structure can equally be characterized by transformations which leave them invariant. From this and under the leadership of the Frenchman Évariste Galois (for algebraic equations) and the Norwegian Sophus Lies (for differential equations and general geometric structures) the 19th century mathematicians developed a new mathematical discipline that build on crystallography (and was demonstrably influenced by it): group theory. It was the transfer from a concrete geometric to an abstract algebraic structure which would make symmetry a “primal principle” of physics.

Probably the most impressive female mathematician of the twentieth century was to finally give the physicists’ striving for symmetry a strict mathematical form: In 1918 Emmy Noether formulated a theorem, which was to be known as the “Noether Theorem”. It combines elementary physical quantities (such as energy, momentum, angular momentum, spin) with algebraic-geometric symmetries, namely the invariance of physics’ fundamental equations under (symmetry) transformations. Thus, for example, the theorem of conservation of energy is based on the property of Newton’s law of classical mechanics (as well as the Schrödinger equation of quantum mechanics as well as any other physical theories accepted today) not to change its form by a shift along the time axis. In other words, from the fact that natural laws do not change from one day to the next results the law of energy conservation. Conversely, in the presence of a conserved quantity the underlying theory must have a certain symmetry. An amazing connection. In modern theoretical physics a symmetry transformations can be much more abstract than simple time shifts. In the abstract spaces of the quantum field theories, for example, the eight gluons of the strong nuclear force, or the existence of two fundamentally different types of quantum particles (bosons and fermions) are the result of some very non-intuitive symmetries. During the 20th century the Noether Theorem became one of the most important foundations of theoretical physics.

The “faith” of scientists is their profound confidence in the beauty in nature in general and the symmetry of its laws in particular. Thereby they prove to be aesthetically sensitive people. Even if the concept of “simple” in modern theoretical physics is somewhat different from the nevertheless everyday understanding and the underlying mathematics is very abstract, its theories and laws are characterized by a beautiful consistency and symmetry. Symmetry is therefore nothing less than the conditio sine qua non of any physical theory. The theoretical physicist, Paul Dirac was the first and possibly most radical to articulate this belief (and implicitly the problems connected to it): “It is more important to have beauty in one’s equations than to have them fit experiment.” The equation he himself derived purely by means of theoretical symmetry considerations – the so-called “Dirac equation” – unified quantum mechanics and special relativity and is today regarded as one of the most impressive expressions of mathematical elegance and beauty in physics. It made numerous astonishing predictions which have all been empirically validated, such as the existence of antimatter. Equally the theoretical physicists postulated the existence of the well-known Higgs particle on the basis of symmetry considerations already in the 1960s. In fact, they were so certain of their theory and thus the existence of this ominous particle that they were willing to wait half a century for its experimental detection (which was announced on July 4, 2012).

Does the physicists’ desire for symmetry possibly have perceptual or motivational psychological origins? Physicists like Dirac certainly did not escape the fact that they render symmetry to something almost metaphysical, in philosophical terminology: to the “very principle of true being”. They thereby drew upon an argumentative structure which bears some strong similarities to medieval scholasticism: Symmetry is true because it is the very principle of existence, and this just because it is beautiful. The question arises whether such “epistemic simplicity” expressed in the physicists’ demand for symmetry in their search for knowledge can be the sole criterion of quality in a scientific theory or even the ideal of science itself. This reveals the entire problem of an attitude such as the one expressed by Dirac: it declares “beautiful” theories a priori as immune and appeals to hold on to them in spite of possible experimental refutation, on the pure basis of their simplicity and symmetry. In the discussion among theoretical physicists on supersymmetry (SUSY) and supersymmetric quantum field theories, this question finds a very current explosiveness. Despite greatest efforts and expenditures the physicists have so far seen absolutely no signs of experimental detection of the so-called SUSY particles associated with it and are therefore forced to find ever new theoretical explanations in order to make their theory compliant with those results. This must remind us of the ongoing alternations in the Ptolemaic model of the universe in medieval times with the aim of making it compatible with more and more contradicting observations.

But anyone who has once grasped how elegant and beautiful a mathematical structure capturing natural processes will hardly fall short of ample amazement. What else but the feeling of indescribable elation Einstein must have felt when he finally realized that his equations of general relativity had the right mathematical properties (the “covariance”) and explained all relevant phenomena of gravity, including the hitherto unexplained such as the precession of the perihelion of Mercury. Heisenberg also describes such a feeling in his autobiography “The Part and the Whole”. He described his feelings at the very moment, when all of a sudden in the signs on the sheet of paper meaning revealed itself to him and he recognized the fundamental laws of the atom:

At the first moment I was deeply shocked. I had the feeling to look through the surface of atomic phenomena down to a ground of strange inner beauty lying deep beneath it and I nearly got dizzy from thinking that I should now go into the matter of investigating this abundance of mathematical structures nature had spread down there in front of me.

However, already Kant showed that, in the satisfaction of human aesthetic needs lies a fundamental difference to our cognitive needs as they express themselves in science. The question of whether the mathematical structures of natural laws with their symmetries have an ontological status independent of ourselves or whether these symmetries are just the condition of the possibility of our experience constitutes the essence of the (late) Kantian philosophy. It has not lost any of its relevance and explosiveness to this day.

To what extent can the striving of modern science (especially physics) for symmetry then be compared with the search for beauty in the arts? In the later there is a clear lack of elements of mathematical symmetry. Here they do not serve as a criterion for beauty or even as an aesthetic ideal. Regular geometric bodies are considered rather uninteresting in paintings or sculptures. Rather, artists consider their works in their uniqueness only through the breaking of symmetry and order. The aesthetic feeling of the scientists is thus determined by an extreme need for order and simplicity as such is hardly found in art. But what about the connections between the arts and science then? If symmetry in the science is more about epistemic motivation than ontological substance, beauty, in the sense of mathematical symmetry, though it constitutes an important scientific criterion of truth, is in the end mainly an important drive and heuristic means of scientific research. As such, it must ultimately be consolidated by experiment and experience. Therefore, both art and science as creative human activities depend on the pursuit of beauty in their own ways. At least this should constitute a bridge between them.

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