# Why does symmetry satisfy people

## Symmetry and Beauty - Bridge Between Science and Art?

There is a term that moves scientists and artists at the same time and that seems like no other to be suitable for building a rare discursive bridge between their two disciplines. It is about the concept of the beautiful, in a narrower context that of "symmetry". Most ancient and modern conceptions of art recognize symmetry as an essential criterion of beauty. Theoretical physics of the 20th century, on the other hand, discovered something like a founding principle in it.

The term is derived from ancient Greek *symmetría* ab, a compound of *syn* (together) and *metron* (the right measure). So symmetry means level or equality. In the ancient conception of art, symmetry described the ideal proportions of length and distance in sculptures (sculpture), pictures (painting) or buildings (architecture). Just like the perfect harmonies in music, these should be expressed in appropriate numerical proportions. The human body was used as a model. For example, the ratio of the length of the arm to the entire body is a quarter. And with outstretched arms and legs, their ends describe exactly a square or a circle with the navel as the center. It is no coincidence that this realization was emphasized during the Renaissance by precisely the scholar who is still considered to be one of the greatest artists and scientists in one person: Leonardo da Vinci. In addition to ideal proportions, ancient art already knew two other concepts for symmetry: mirror symmetry, as it is expressed in the relationship between the left and right halves of the body, and the balance of opposites, as found, for example, in Greek medicine and its doctrine of the Body fluids articulated.

The essential endeavor of natural scientists, on the other hand, is to work out simple processes and structures from the confusing complexity of natural phenomena. Most theoretical physicists cherish the deep belief that nature, in spite of the diversity of its phenomena, proves to be simple in its structure on a fundamental level. And in this simplicity, which is reflected in the mathematical structures of the physical theories that describe them, they want to recognize the true beauty of nature. That beauty expresses its expression precisely through the discipline, the grammar of which many artists have never particularly appreciated or mastered, may seem ironic. For a physicist, “simple” is pretty much anything that can be represented mathematically precisely. So it is not easy what nature immediately offers us, but first the scientists have to separate the colorful mixture of phenomena, free the important from all unnecessary accessories (such as the friction during free fall) until the "simple “Show events. Only this simple thing can then appear “beautiful”.

This separation of “unnecessary accessories” is easiest in astronomy (there is no friction in space), which is why this also represented the starting point of the scientific revolution. Johannes Kepler was so enthusiastic about the beauty and simplicity of the heavenly movements, which were revealed to him in his laws of planetary movement, that he saw in them the highest divine principles. And Newton not only provided physics with the mathematics with which the planetary motion can be calculated concretely, but he also gave the view, still quite daring expressed by Galileo, that “the book of nature is written in the language of mathematics”, its real one Eligibility (and science with it the claim to be able to derive and calculate all natural occurrences, a claim which it maintains in one form or another to this day and which should make it the most influential social and intellectual force of the modern age). Einstein was no less moved by the stringency and sublimity expressed in the mathematical structure of the laws of nature. His general theory of relativity, which brought the phenomenon of gravity into a wonderful geometric formulation, is still considered one of the most uplifting and beautiful theories in nature to this day. Heisenberg agrees: “Similar to Plato, the final theory of matter will be characterized by a number of important symmetry requirements.” But these symmetries are no longer necessarily clear, as he continues: “These symmetries can no longer be seen simply explain them using figures and pictures, as was possible with platonic solids, but using equations. "

As Heisenberg already suggests, the concept of symmetry in the natural sciences is different from that in art. This is less about proportions or equilibrium than about order and structure. In the mathematical definition of nature, symmetry describes an important aspect in the characterization of the structure and dynamics of natural objects. One example is the classification of crystals as it began in the 18th century and ended in the 19th century. The symmetries of crystal forms are shown in the fact that rotations around certain angles (and axes) do not change their appearance. As already stated by Kepler, snow crystals are always symmetrical like a hexagon for all their individuality (the reason is the special shape of the water molecule). The influence of crystallography was also clearly evident when a generalized mathematical concept of symmetry was developed in the 19th century: invariance to transformations. A structure is considered to be symmetrical if certain rotations convert it back into the same shape.

Just like the rotations of bodies, algebraic equations, differential equations and general geometric structures can also be characterized by transformations that leave them invariant. From this, the mathematicians in the 19th century, under the leadership of the French Évariste Galois (for algebraic equations) and the Norwegian Sophus Lies (for differential equations and general geometric structures), developed a completely new mathematical discipline that followed the scheme of crystallography (and demonstrably influenced by it war): the group theory. It was the transfer from a concrete geometric structure to an abstract algebraic structure that was supposed to make symmetry a 'primal principle' of physics.

Probably the most impressive mathematician of the 20th century was ultimately to give the physicists' striving for symmetry a stringent mathematical form: In 1918 Emmy Noether formulated a theorem that would become known as the “Noether Theorem”. It links elementary physical quantities (such as energy, momentum, angular momentum, charge, spin) with algebraic-geometric symmetries, namely the invariance of basic physical equations under certain (symmetry) transformations. For example, the law of energy conservation is based on the property of Newton's law of classical mechanics (as well as the Schrödinger equation of quantum mechanics and all other physical equations accepted today) that it does not change its shape when it is shifted on the time axis. In other words: The law of energy conservation results from the fact that the laws of nature do not change overnight. Conversely, if a conserved quantity is available, the underlying theory must have a certain symmetry. An amazing connection. In modern theoretical physics, symmetry transformations can turn out to be much more abstract than simple temporal shifts. There are very few clear symmetries in the abstract spaces of quantum field theories, such as the eight gluons of the strong nuclear force or the existence of two fundamentally different types of quantum particles (bosons and fermions). The Noether theorem became one of the most important foundations of theoretical physics in the course of the 20th century.

The "belief" of scientists is a deep trust in the beauty of nature in general and the symmetry of its laws in particular. In doing so, they prove to be aesthetically sensitive people. Even if the concept of "simple" in modern theoretical physics is somewhat different from colloquial and the mathematics on which it is based is highly abstract, its theories and laws are nonetheless characterized by a wonderful consistency and symmetry. So symmetry is nothing less than that *conditio sine qua non* of any physical theory. The theoretical physicist Paul Dirac was the first and probably the most radical to articulate this belief (and inadvertently also the related problem): "It is more important to have beauty in your equations than to agree with the experiment". The combination of quantum mechanics and special relativity theory in the so-called “Dirac equation”, which he derived on the basis of purely theoretical symmetry considerations, is still considered to be one of the most impressive examples of mathematical elegance and beauty in physics. Numerous astonishing and now without exception empirically validated predictions follow from it, such as the existence of antimatter. And the existence of the well-known Higgs particle was postulated by physicists on the basis of symmetry considerations as early as the 1960s. They were so sure of their theory and thus of the existence of this ominous particle that they were ready to wait half a century for its experimental detection (which was finally announced on July 4th, 2012).

But doesn't physicists' desire for symmetry also have origins in perception or in motivational psychology? Physicists like Dirac certainly did not miss the fact that they stylized symmetry into something almost metaphysical, in philosophical terminology: a “principle of true beings”. In doing so, they rely on a set of arguments that are quite similar to that of medieval scholasticism: symmetry is true because it is the principle of being, and this simply because it is beautiful. The question that arises here involuntarily is whether such “epistemic simplicity”, as it is expressed in the symmetry requirement of the physicists striving for knowledge, can be the sole criterion of a scientific theory or even the ideal of science. This reveals the whole problem of an attitude as it finds its expression in Dirac's words: It declares "beautiful" theories a priori as immune and appeals to hold on to them despite possible experimental refutation, solely because of their simplicity and symmetry. In today's discussion among theoretical physicists about supersymmetry (SUSY) and supersymmetric quantum field theories, this question is very hot. Because despite the greatest efforts and expenditures, the physicists have not found the slightest sign of experimental evidence for the so-called SUSY particles associated with them. They always have to come up with new explanations to reconcile these zero results with their theory. The whole thing is more reminiscent of the ongoing messing around in the Ptolemaic worldview in the Middle Ages, in order to bring this into harmony with the increasingly numerous contradicting observations that have become more and more numerous over time.

But once you have grasped how elegant and almost wonderful a mathematical structure can be when grasping nature, you can hardly stop being amazed. Einstein must have felt something other than the feeling of indescribable elation when he last noticed that his equations of general relativity had the correct mathematical properties (the "covariance") and at the same time all the phenomena related to gravitation, including those previously unexplained such as perihelion - Rotation of Mercury, describe exactly. Heisenberg describes such a feeling in his autobiography “The part and the whole”. He was describing his feelings at the very moment when the signs on the piece of paper revealed meaning to him and he recognized the basic laws of atoms:

At first I was deeply shocked. I had the feeling that I was looking through the surface of the atomic phenomena to a deep underlying ground of strange inner beauty, and it almost made me dizzy to think that I should now pursue this plethora of mathematical structures that nature there had spread out below in front of me.

But even Kant showed that the satisfaction of human aesthetic needs is a fundamental difference to our needs for knowledge, as they find their expression in science. The question of whether the mathematical structures of the laws of nature with their symmetries have an ontological status that is independent of us, or whether these symmetries only represent the condition of the possibility of our experience, determines the entire (late) Kantian philosophy. To this day it has lost none of its relevance and explosiveness.

To what extent can the striving of modern science (especially physics) for symmetry be compared with the search for beauty in art? In the fine arts there is a lack of mathematical symmetry elements. Here they hardly serve as a criterion for beauty or even as an aesthetic ideal. Regular geometric bodies in pictures and sculptures tend to be considered uninteresting. Rather, artists understand their works in their uniqueness only through symmetry and order*breaks*. The aesthetic perception of scientists is thus determined by an extreme need for order and simplicity, which is hardly matched in art. But what about the connections between art and science? If symmetries in science are more about epistemic motivation than about ontological substance, then beauty in the sense of mathematical symmetry represents an important scientific truth criterion, but is mainly an important source of motivation and a heuristic means of scientific research. As such, it must ultimately be measured by experiment and experience. Hence, both art and science, as creative human activities, each rely in their own way on the pursuit of beauty. At least this should be some kind of bridge between them.

Born in 1969, I studied physics and philosophy at the University of Bonn and the École Polytechnique in Paris in the 1990s, before doing my doctorate in theoretical physics at the Max Planck Institute for the Physics of Complex Systems in Dresden, where I also did my post- Doc studies did further research in the field of nonlinear dynamics. Before that, I had also worked in the field of quantum field theories and particle physics. Meanwhile, I've been living in Switzerland for almost 20 years. For many years I have dealt with border issues in modern (as well as historical) sciences. In my books, blogs, and articles, I focus on the subjects of science, philosophy, and spirituality, especially the history of science, its relationship to spiritual traditions, and its impact on modern society. In the past I have also written on investment topics (alternative investments). My two books “Naturwissenschaft: Eine Biographie” and “Wissenschaft und Spiritualität” were published by Springer Spektrum Verlag in 2015 and 2016. I have been running my blog since 2014 at www.larsjaeger.ch.

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